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== 8.8#h(0.6em)Use the Complex Number System

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Given the numbers #math.equation(block: false, alt: "−4 , minus the square root of 7 , 0. 5 minus , the fraction 7 over 3 , 3 , the square root of 81 ,")[$−4 , − sqrt(7) , 0. accent(5, −) , frac(7, 3) , 3 , sqrt(81) ,$] list the ⓐ  rational numbers, ⓑ irrational numbers, ⓒ real numbers. #linebreak() 
If you missed this problem, review .

#solutionbox[
ⓐ #math.equation(block: false, alt: "−4 , 0. 5 minus , the fraction 7 over 3 , 3 , the square root of 81 ;")[$−4 , 0. accent(5, −) , frac(7, 3) , 3 , sqrt(81) ;$] ⓑ #math.equation(block: false, alt: "the square root of 7 ;")[$sqrt(7) ;$] ⓒ #math.equation(block: false, alt: "−4 , minus the square root of 7 , 0. 5 minus , the fraction 7 over 3 , 3 , the square root of 81")[$−4 , − sqrt(7) , 0. accent(5, −) , frac(7, 3) , 3 , sqrt(81)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis x minus 3 close parenthesis open parenthesis 2 x plus 5 close parenthesis .")[$( x − 3 ) ( 2 x + 5 ) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "2 x squared minus x minus 15")[$2 x^(2) − x − 15$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Rationalize the denominator:#math.equation(block: false, alt: "the fraction the square root of 5 over the square root of 5 minus the square root of 3 .")[$frac(sqrt(5), sqrt(5) − sqrt(3)) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 plus the square root of 15 over 2")[$frac(5 + sqrt(15), 2)$]
]
]


=== Evaluate the Square Root of a Negative Number

Whenever we have a situation where we have a square root of a negative number we say there is no real number that equals that square root. For example, to simplify #math.equation(block: false, alt: "the square root of −1 ,")[$sqrt(−1) ,$] we are looking for a real number #emph[x] so that #emph[x]#super[2] = –1. Since all real numbers squared are positive numbers, there is no real number that equals –1 when squared.

Mathematicians have often expanded their numbers systems as needed. They added 0 to the counting numbers to get the whole numbers. When they needed negative balances, they added negative numbers to get the integers. When they needed the idea of parts of a whole they added fractions and got the rational numbers. Adding the irrational numbers allowed numbers like #math.equation(block: false, alt: "the square root of 5 .")[$sqrt(5) .$] All of these together gave us the real numbers and so far in your study of mathematics, that has been sufficient.

But now we will expand the real numbers to include the square roots of negative numbers. We start by defining the #strong[imaginary unit] #math.equation(block: false, alt: "i")[$i$] as the number whose square is –1.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Imaginary Unit]

The #strong[imaginary unit] #emph[i] is the number whose square is –1.

#math.equation(block: true, alt: "i squared equals −1 or i equals the square root of −1")[$i^(2) = −1 #h(0.2em) "or" #h(0.2em) i = sqrt(−1)$]
]

We will use the imaginary unit to simplify the square roots of negative numbers.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Square Root of a Negative Number]

If #emph[b] is a positive real number, then

#math.equation(block: true, alt: "the square root of − b equals the square root of b i")[$sqrt("−" b) = sqrt(b) #h(0.2em) i$]
]

We will use this definition in the next example. Be careful that it is clear that the #emph[i] is not under the radical. Sometimes you will see this written as #math.equation(block: false, alt: "the square root of − b equals i the square root of b")[$sqrt("−" b) = i sqrt(b)$] to emphasize the #emph[i] is not under the radical. But the #math.equation(block: false, alt: "the square root of − b equals the square root of b i")[$sqrt("−" b) = sqrt(b) #h(0.2em) i$] is considered standard form.

#examplebox("Example 1")[][
Write each expression in terms of #emph[i] and simplify if possible:

ⓐ #math.equation(block: false, alt: "the square root of −25")[$sqrt(−25)$] ⓑ #math.equation(block: false, alt: "the square root of −7")[$sqrt(−7)$] ⓒ #math.equation(block: false, alt: "the square root of −12 .")[$sqrt(−12) .$]

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of −25")[$sqrt(−25)$]]),
  [Use the definition of the square root of negative numbers.], [#math.equation(block: false, alt: "the square root of 25 i")[$sqrt(25) #h(0.2em) i$]],
  [Simplify.], [#math.equation(block: false, alt: "5 i")[$5 i$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of −7")[$sqrt(−7)$]]),
  [Use the definition of the square root of negative numbers.], [#math.equation(block: false, alt: "the square root of 7 i")[$sqrt(7) i$]],
  [Simplify.], [Be careful that it is clear that #math.equation(block: false, alt: "i")[$i$] is not under the radical sign.],
))

ⓒ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of −12")[$sqrt(−12)$]]),
  [Use the definition of the square root of negative numbers.], [#math.equation(block: false, alt: "the square root of 12 i")[$sqrt(12) #h(0.2em) i$]],
  [Simplify #math.equation(block: false, alt: "the square root of 12 .")[$sqrt(12) .$]], [#math.equation(block: false, alt: "2 the square root of 3 i")[$2 sqrt(3) #h(0.2em) i$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write each expression in terms of #emph[i] and simplify if possible:

ⓐ #math.equation(block: false, alt: "the square root of −81")[$sqrt(−81)$] ⓑ #math.equation(block: false, alt: "the square root of −5")[$sqrt(−5)$] ⓒ #math.equation(block: false, alt: "the square root of −18 .")[$sqrt(−18) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "9 i")[$9 i$] ⓑ #math.equation(block: false, alt: "the square root of 5 i")[$sqrt(5) i$] ⓒ #math.equation(block: false, alt: "3 the square root of 2 i")[$3 sqrt(2) i$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write each expression in terms of #emph[i] and simplify if possible:

ⓐ #math.equation(block: false, alt: "the square root of −36")[$sqrt(−36)$] ⓑ #math.equation(block: false, alt: "the square root of −3")[$sqrt(−3)$] ⓒ #math.equation(block: false, alt: "the square root of −27 .")[$sqrt(−27) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "6 i")[$6 i$] ⓑ #math.equation(block: false, alt: "the square root of 3 i")[$sqrt(3) i$] ⓒ #math.equation(block: false, alt: "3 the square root of 3 i")[$3 sqrt(3) i$]
]
]

Now that we are familiar with the imaginary number #emph[i], we can expand our concept of the number system to include imaginary numbers. The #strong[complex number system] includes the real numbers and the imaginary numbers. A #strong[complex number] is of the form #emph[a] + #emph[bi], where #emph[a, b] are real numbers. We call #emph[a] the real part and #emph[b] the imaginary part.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Complex Number]

A #strong[complex number] is of the form #emph[a] + #emph[bi], where #emph[a] and #emph[b] are real numbers.

#figure(figph[The image shows the expression a plus b i. The number a is labeled “real part” and the number b i is labeled “imaginary part”.], alt: "The image shows the expression a plus b i. The number a is labeled “real part” and the number b i is labeled “imaginary part”.", caption: none)
]

A complex number is in standard form when written as #math.equation(block: false, alt: "a plus b i ,")[$a + b i ,$] where #emph[a] and #emph[b] are real numbers.

If #math.equation(block: false, alt: "b equals 0 ,")[$b = 0 ,$] then #math.equation(block: false, alt: "a plus b i")[$a + b i$] becomes #math.equation(block: false, alt: "a plus 0 times i equals a ,")[$a + 0 · i = a ,$] and is a real number.

If #math.equation(block: false, alt: "b not equal to 0 ,")[$b ≠ 0 ,$] then #math.equation(block: false, alt: "a plus b i")[$a + b i$] is an imaginary number.

If #math.equation(block: false, alt: "a equals 0 ,")[$a = 0 ,$] then #math.equation(block: false, alt: "a plus b i")[$a + b i$] becomes #math.equation(block: false, alt: "0 plus b i equals b i ,")[$0 + b i = b i ,$] and is called a pure imaginary number.

We summarize this here.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "a plus b i")[$a + b i$]], []),
  [#math.equation(block: false, alt: "b equals 0")[$b = 0$]], [#math.equation(block: false, alt: "a plus 0 times i; a")[$a + 0 · i \  \ #h(1.2em) a$]], [Real number],
  [#math.equation(block: false, alt: "b not equal to 0")[$b ≠ 0$]], [#math.equation(block: false, alt: "a plus b i")[$a + b i$]], [Imaginary number],
  [#math.equation(block: false, alt: "a equals 0")[$a = 0$]], [#math.equation(block: false, alt: "0 plus b i; b i")[$0 + b i \  \  \ #h(0.8em) b i$]], [Pure imaginary number],
))

The standard form of a complex number is #math.equation(block: false, alt: "a plus b i ,")[$a + b i ,$] so this explains why the preferred form is #math.equation(block: false, alt: "the square root of − b equals the square root of b i")[$sqrt("−" b) = sqrt(b) i$] when #math.equation(block: false, alt: "b greater than 0 .")[$b > 0 .$]

The diagram helps us visualize the complex number system. It is made up of both the real numbers and the imaginary numbers.

#figure(figph[The table has four rows and three columns. The first row is a header and the second column entry a plus b i. In the second row is b equals zero, a plus 0 i, and “Real number”. The third row contains b is not equal to 0, a plus b i, and “Imaginary number”. The fourth row contains a = 0, 0 plus b i, and “Pure imaginary number”.], alt: "The table has four rows and three columns. The first row is a header and the second column entry a plus b i. In the second row is b equals zero, a plus 0 i, and “Real number”. The third row contains b is not equal to 0, a plus b i, and “Imaginary number”. The fourth row contains a = 0, 0 plus b i, and “Pure imaginary number”.", caption: none)


=== Add or Subtract Complex Numbers

We are now ready to perform the operations of addition, subtraction, multiplication and division on the complex numbers—just as we did with the real numbers.

Adding and subtracting complex numbers is much like adding or subtracting like terms. We add or subtract the real parts and then add or subtract the imaginary parts. Our final result should be in standard form.

#examplebox("Example 2")[][
Add: #math.equation(block: false, alt: "the square root of −12 plus the square root of −27 .")[$sqrt(−12) + sqrt(−27) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of −12 plus the square root of −27")[$sqrt(−12) + sqrt(−27)$]]),
  [Use the definition of the square root of negative numbers.], [#math.equation(block: false, alt: "the square root of 12 i plus the square root of 27 i")[$sqrt(12) #h(0.2em) i + sqrt(27) #h(0.2em) i$]],
  [Simplify the square roots.], [#math.equation(block: false, alt: "2 the square root of 3 i plus 3 the square root of 3 i")[$2 sqrt(3) #h(0.2em) i + 3 sqrt(3) #h(0.2em) i$]],
  [Add.], [#math.equation(block: false, alt: "5 the square root of 3 i")[$5 sqrt(3) #h(0.2em) i$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "the square root of −8 plus the square root of −32 .")[$sqrt(−8) + sqrt(−32) .$]

#solutionbox[
#math.equation(block: true, alt: "6 the square root of 2 i")[$6 sqrt(2) i$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "the square root of −27 plus the square root of −48 .")[$sqrt(−27) + sqrt(−48) .$]

#solutionbox[
#math.equation(block: true, alt: "7 the square root of 3 i")[$7 sqrt(3) i$]
]
]

Remember to add both the real parts and the imaginary parts in this next example.

#examplebox("Example 3")[][
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis 4 minus 3 i close parenthesis plus open parenthesis 5 plus 6 i close parenthesis")[$( 4 − 3 i ) + ( 5 + 6 i )$] ⓑ #math.equation(block: false, alt: "open parenthesis 2 minus 5 i close parenthesis minus open parenthesis 5 minus 2 i close parenthesis .")[$( 2 − 5 i ) − ( 5 − 2 i ) .$]

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 4 minus 3 i close parenthesis plus open parenthesis 5 plus 6 i close parenthesis")[$( 4 − 3 i ) + ( 5 + 6 i )$]]),
  [Use the Associative Property to put the real #linebreak() parts and the imaginary parts together.], [#math.equation(block: false, alt: "open parenthesis 4 plus 5 close parenthesis plus open parenthesis −3 i plus 6 i close parenthesis")[$( 4 + 5 ) + ( −3 i + 6 i )$]],
  [Simplify.], [#math.equation(block: false, alt: "9 plus 3 i")[$9 + 3 i$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 2 minus 5 i close parenthesis minus open parenthesis 5 minus 2 i close parenthesis")[$( 2 − 5 i ) − ( 5 − 2 i )$]]),
  [Distribute.], [#math.equation(block: false, alt: "2 minus 5 i minus 5 plus 2 i")[$2 − 5 i − 5 + 2 i$]],
  [Use the Associative Property to put the real #linebreak() parts and the imaginary parts together.], [#math.equation(block: false, alt: "2 minus 5 minus 5 i plus 2 i")[$2 − 5 − 5 i + 2 i$]],
  [Simplify.], [#math.equation(block: false, alt: "−3 minus 3 i")[$−3 − 3 i$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis 2 plus 7 i close parenthesis plus open parenthesis 4 minus 2 i close parenthesis")[$( 2 + 7 i ) + ( 4 − 2 i )$] ⓑ #math.equation(block: false, alt: "open parenthesis 8 minus 4 i close parenthesis minus open parenthesis 2 minus i close parenthesis .")[$( 8 − 4 i ) − ( 2 − i ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "6 plus 5 i")[$6 + 5 i$] ⓑ #math.equation(block: false, alt: "6 minus 3 i")[$6 − 3 i$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis 3 minus 2 i close parenthesis plus open parenthesis −5 minus 4 i close parenthesis")[$( 3 − 2 i ) + ( −5 − 4 i )$] ⓑ #math.equation(block: false, alt: "open parenthesis 4 plus 3 i close parenthesis minus open parenthesis 2 minus 6 i close parenthesis .")[$( 4 + 3 i ) − ( 2 − 6 i ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−2 minus 6 i")[$−2 − 6 i$] ⓑ #math.equation(block: false, alt: "2 plus 9 i")[$2 + 9 i$]
]
]


=== Multiply Complex Numbers

Multiplying complex numbers is also much like multiplying expressions with coefficients and variables. There is only one special case we need to consider. We will look at that after we practice in the next two examples.

#examplebox("Example 4")[][
Multiply: #math.equation(block: false, alt: "2 i open parenthesis 7 minus 5 i close parenthesis .")[$2 i ( 7 − 5 i ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "2 i open parenthesis 7 minus 5 i close parenthesis")[$2 i ( 7 − 5 i )$]]),
  [Distribute.], [#math.equation(block: false, alt: "14 i minus 10 i squared")[$14 i − 10 i^(2)$]],
  [Simplify #math.equation(block: false, alt: "i squared .")[$i^(2) .$]], [#math.equation(block: false, alt: "14 i minus 10 open parenthesis −1 close parenthesis")[$14 i − 10 ( −1 )$]],
  [Multiply.], [#math.equation(block: false, alt: "14 i plus 10")[$14 i + 10$]],
  [Write in standard form.], [#math.equation(block: false, alt: "10 plus 14 i")[$10 + 14 i$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "4 i open parenthesis 5 minus 3 i close parenthesis .")[$4 i ( 5 − 3 i ) .$]

#solutionbox[
#math.equation(block: true, alt: "12 plus 20 i")[$12 + 20 i$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "−3 i open parenthesis 2 plus 4 i close parenthesis .")[$−3 i ( 2 + 4 i ) .$]

#solutionbox[
#math.equation(block: true, alt: "12 minus 6 i")[$12 − 6 i$]
]
]

In the next example, we multiply the binomials using the #strong[Distributive Property] or #strong[FOIL].

#examplebox("Example 5")[][
Multiply: #math.equation(block: false, alt: "open parenthesis 3 plus 2 i close parenthesis open parenthesis 4 minus 3 i close parenthesis .")[$( 3 + 2 i ) ( 4 − 3 i ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 3 plus 2 i close parenthesis open parenthesis 4 minus 3 i close parenthesis")[$( 3 + 2 i ) ( 4 − 3 i )$]]),
  [Use FOIL.], [#math.equation(block: false, alt: "12 minus 9 i plus 8 i minus 6 i squared")[$12 − 9 i + 8 i − 6 i^(2)$]],
  [Simplify #math.equation(block: false, alt: "i squared")[$i^(2)$] and combine like terms.], [#math.equation(block: false, alt: "12 minus i minus 6 open parenthesis −1 close parenthesis")[$12 − i − 6 ( −1 )$]],
  [Multiply.], [#math.equation(block: false, alt: "12 minus i plus 6")[$12 − i + 6$]],
  [Combine the real parts.], [#math.equation(block: false, alt: "18 minus i")[$18 − i$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 5 minus 3 i close parenthesis open parenthesis −1 minus 2 i close parenthesis .")[$( 5 − 3 i ) ( −1 − 2 i ) .$]

#solutionbox[
#math.equation(block: true, alt: "−11 minus 7 i")[$−11 − 7 i$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis −4 minus 3 i close parenthesis open parenthesis 2 plus i close parenthesis .")[$( −4 − 3 i ) ( 2 + i ) .$]

#solutionbox[
#math.equation(block: true, alt: "−5 minus 10 i")[$−5 − 10 i$]
]
]

In the next example, we could use FOIL or the #strong[Product of Binomial Squares Pattern].

#examplebox("Example 6")[][
Multiply: #math.equation(block: false, alt: "open parenthesis 3 plus 2 i close parenthesis squared")[$attach(( 3 + 2 i ), t: 2)$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing the fraction (a + b) / (3 + 2i) squared. The numerator 'a + b' is displayed in red text.], alt: "A mathematical expression showing the fraction (a + b) / (3 + 2i) squared. The numerator 'a + b' is displayed in red text.", caption: none)]),
  [Use the Product of Binomial Squares Pattern, #math.equation(block: false, alt: "open parenthesis a plus b close parenthesis squared equals a squared plus 2 a b plus b squared .")[$attach(( a + b ), t: 2) = a^(2) + 2 a b + b^(2) .$]], [#figure(figph[An algebraic identity a^2 + 2ab + b^2 is shown with a specific example: 3^2 + 2 \* 3 \* 2i + (2i)^2, demonstrating the expansion of (a+b)^2 where a=3 and b=2i.], alt: "An algebraic identity a^2 + 2ab + b^2 is shown with a specific example: 3^2 + 2 * 3 * 2i + (2i)^2, demonstrating the expansion of (a+b)^2 where a=3 and b=2i.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical expression 9 + 12i + 4i^2 is shown.], alt: "The mathematical expression 9 + 12i + 4i^2 is shown.", caption: none)],
  [Simplify #math.equation(block: false, alt: "i squared .")[$i^(2) .$]], [#figure(figph[A mathematical expression showing the sum of 9, 12i, and 4 multiplied by -1, which is 9 + 12i + 4(-1).], alt: "A mathematical expression showing the sum of 9, 12i, and 4 multiplied by -1, which is 9 + 12i + 4(-1).", caption: none)],
  [Simplify.], [#figure(figph[The image shows the complex number expression 5 + 12i in black text against a white background.], alt: "The image shows the complex number expression 5 + 12i in black text against a white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply using the Binomial Squares pattern: #math.equation(block: false, alt: "open parenthesis −2 minus 5 i close parenthesis squared .")[$attach(( −2 − 5 i ), t: 2) .$]

#solutionbox[
#math.equation(block: true, alt: "−21 plus 20 i")[$−21 + 20 i$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply using the Binomial Squares pattern: #math.equation(block: false, alt: "open parenthesis −5 plus 4 i close parenthesis squared .")[$attach(( −5 + 4 i ), t: 2) .$]

#solutionbox[
#math.equation(block: true, alt: "9 minus 40 i")[$9 − 40 i$]
]
]

Since the square root of a negative number is not a real number, when we have the square roots of two negative numbers, we cannot use the Product Property for Radicals. In order to multiply square roots of negative numbers we should first write them as complex numbers, using #math.equation(block: false, alt: "the square root of − b equals the square root of b i .")[$sqrt("−" b) = sqrt(b) i .$] This is one place students tend to make errors, so be careful when you see multiplying with a negative square root.

#examplebox("Example 7")[][
Multiply: #math.equation(block: false, alt: "the square root of −36 times the square root of −4 .")[$sqrt(−36) · sqrt(−4) .$]

#solutionbox[
To multiply square roots of negative numbers, we first write them as complex numbers.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of −36 times the square root of −4")[$sqrt(−36) · sqrt(−4)$]]),
  [Write as complex numbers using #math.equation(block: false, alt: "the square root of − b equals the square root of b i .")[$sqrt("−" b) = sqrt(b) i .$]], [#math.equation(block: false, alt: "the square root of 36 i times the square root of 4 i")[$sqrt(36) #h(0.2em) i · sqrt(4) #h(0.2em) i$]],
  [Simplify.], [#math.equation(block: false, alt: "6 i times 2 i")[$6 i · 2 i$]],
  [Multiply.], [#math.equation(block: false, alt: "12 i squared")[$12 i^(2)$]],
  [Simplify #math.equation(block: false, alt: "i squared")[$i^(2)$] and multiply.], [#math.equation(block: false, alt: "−12")[$−12$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "the square root of −49 times the square root of −4 .")[$sqrt(−49) · sqrt(−4) .$]

#solutionbox[
#math.equation(block: true, alt: "−14")[$−14$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "the square root of −36 times the square root of −81 .")[$sqrt(−36) · sqrt(−81) .$]

#solutionbox[
#math.equation(block: true, alt: "−54")[$−54$]
]
]

In the next example, each binomial has a square root of a negative number. Before multiplying, each square root of a negative number must be written as a complex number.

#examplebox("Example 8")[][
Multiply: #math.equation(block: false, alt: "open parenthesis 3 minus the square root of −12 close parenthesis open parenthesis 5 plus the square root of −27 close parenthesis .")[$( 3 − sqrt(−12) ) ( 5 + sqrt(−27) ) .$]

#solutionbox[
To multiply square roots of negative numbers, we first write them as complex numbers.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 3 minus the square root of −12 close parenthesis open parenthesis 5 plus the square root of −27 close parenthesis")[$( 3 − sqrt(−12) ) ( 5 + sqrt(−27) )$]]),
  [Write as complex numbers using #math.equation(block: false, alt: "the square root of − b equals the square root of b i .")[$sqrt("−" b) = sqrt(b) i .$]], [#math.equation(block: false, alt: "open parenthesis 3 minus 2 the square root of 3 i close parenthesis open parenthesis 5 plus 3 the square root of 3 i close parenthesis")[$( 3 − 2 sqrt(3) #h(0.2em) i ) ( 5 + 3 sqrt(3) #h(0.2em) i )$]],
  [Use FOIL.], [#math.equation(block: false, alt: "15 plus 9 the square root of 3 i minus 10 the square root of 3 i minus 6 times 3 i squared")[$15 + 9 sqrt(3) #h(0.2em) i − 10 sqrt(3) #h(0.2em) i − 6 · 3 i^(2)$]],
  [Combine like terms and simplify #math.equation(block: false, alt: "i squared .")[$i^(2) .$]], [#math.equation(block: false, alt: "15 minus the square root of 3 i minus 6 times open parenthesis −3 close parenthesis")[$15 − sqrt(3) #h(0.2em) i − 6 · ( −3 )$]],
  [Multiply and combine like terms.], [#math.equation(block: false, alt: "33 minus the square root of 3 i")[$33 − sqrt(3) #h(0.2em) i$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 4 minus the square root of −12 close parenthesis open parenthesis 3 minus the square root of −48 close parenthesis .")[$( 4 − sqrt(−12) ) ( 3 − sqrt(−48) ) .$]

#solutionbox[
#math.equation(block: true, alt: "−12 minus 22 the square root of 3 i")[$−12 − 22 sqrt(3) i$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis −2 plus the square root of −8 close parenthesis open parenthesis 3 minus the square root of −18 close parenthesis .")[$( −2 + sqrt(−8) ) ( 3 − sqrt(−18) ) .$]

#solutionbox[
#math.equation(block: true, alt: "6 plus 12 the square root of 2 i")[$6 + 12 sqrt(2) i$]
]
]

We first looked at conjugate pairs when we studied polynomials. We said that a pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference is called a #emph[conjugate pair] and is of the form #math.equation(block: false, alt: "open parenthesis a minus b close parenthesis , open parenthesis a plus b close parenthesis .")[$( a − b ) , ( a + b ) .$]

A #strong[complex conjugate pair] is very similar. For a complex number of the form #math.equation(block: false, alt: "a plus b i ,")[$a + b i ,$] its conjugate is #math.equation(block: false, alt: "a minus b i .")[$a − b i .$] Notice they have the same first term and the same last term, but one is a sum and one is a difference.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Complex Conjugate Pair]

A #strong[complex conjugate pair] is of the form #math.equation(block: false, alt: "a plus b i ,")[$a + b i ,$]#math.equation(block: false, alt: "a minus b i .")[$a − b i .$]
]

We will multiply a complex conjugate pair in the next example.

#examplebox("Example 9")[][
Multiply: #math.equation(block: false, alt: "open parenthesis 3 minus 2 i close parenthesis open parenthesis 3 plus 2 i close parenthesis .")[$( 3 − 2 i ) ( 3 + 2 i ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 3 minus 2 i close parenthesis open parenthesis 3 plus 2 i close parenthesis")[$( 3 − 2 i ) ( 3 + 2 i )$]]),
  [Use FOIL.], [#math.equation(block: false, alt: "9 plus 6 i minus 6 i minus 4 i squared")[$9 + 6 i − 6 i − 4 i^(2)$]],
  [Combine like terms and simplify #math.equation(block: false, alt: "i squared .")[$i^(2) .$]], [#math.equation(block: false, alt: "9 minus 4 open parenthesis −1 close parenthesis")[$9 − 4 ( −1 )$]],
  [Multiply and combine like terms.], [13],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 4 minus 3 i close parenthesis times open parenthesis 4 plus 3 i close parenthesis .")[$( 4 − 3 i ) · ( 4 + 3 i ) .$]

#solutionbox[
25
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis −2 plus 5 i close parenthesis times open parenthesis −2 minus 5 i close parenthesis .")[$( −2 + 5 i ) · ( −2 − 5 i ) .$]

#solutionbox[
29
]
]

From our study of polynomials, we know the product of conjugates is always of the form #math.equation(block: false, alt: "open parenthesis a minus b close parenthesis open parenthesis a plus b close parenthesis equals a squared minus b squared .")[$( a − b ) ( a + b ) = a^(2) − b^(2) .$] The result is called a #strong[difference of squares]. We can multiply a complex conjugate pair using this pattern.

The last example we used FOIL. Now we will use the #strong[Product of Conjugates Pattern].

#figure(figph[The quantity a minus b in parentheses times the quantity a plus b in parentheses is written above the expression showing the product of 3 minus 2 i in parentheses and 3 plus 2 i in parentheses. In the next line a squared minus b squared is written above the expression 3 squared minus the quantity 2 i in parentheses squared. Simplifying we get 9 minus 4 i squared. This is equal to 9 minus 4 times negative 1. The final result is 13.], alt: "The quantity a minus b in parentheses times the quantity a plus b in parentheses is written above the expression showing the product of 3 minus 2 i in parentheses and 3 plus 2 i in parentheses. In the next line a squared minus b squared is written above the expression 3 squared minus the quantity 2 i in parentheses squared. Simplifying we get 9 minus 4 i squared. This is equal to 9 minus 4 times negative 1. The final result is 13.", caption: none)

Notice this is the same result we found in .

When we multiply complex conjugates, the product of the last terms will always have an #math.equation(block: false, alt: "i squared")[$i^(2)$] which simplifies to #math.equation(block: false, alt: "−1 .")[$−1 .$]

#math.equation(block: true, alt: "open parenthesis a minus b i close parenthesis open parenthesis a plus b i close parenthesis; a squared minus open parenthesis b i close parenthesis squared; a squared minus b squared i squared; a squared minus b squared open parenthesis −1 close parenthesis; a squared plus b squared")[$( a − b i ) ( a + b i ) \ a^(2) − attach(( b i ), t: 2) \ a^(2) − b^(2) i^(2) \ a^(2) − b^(2) ( −1 ) \ a^(2) + b^(2)$]This leads us to the Product of Complex Conjugates Pattern: #math.equation(block: false, alt: "open parenthesis a minus b i close parenthesis open parenthesis a plus b i close parenthesis equals a squared plus b squared")[$( a − b i ) ( a + b i ) = a^(2) + b^(2)$]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Product of Complex Conjugates]

If #emph[a] and #emph[b] are real numbers, then

#math.equation(block: true, alt: "open parenthesis a minus b i close parenthesis open parenthesis a plus b i close parenthesis equals a squared plus b squared")[$( a − b i ) ( a + b i ) = a^(2) + b^(2)$]
]

#examplebox("Example 10")[][
Multiply using the Product of Complex Conjugates Pattern: #math.equation(block: false, alt: "open parenthesis 8 minus 2 i close parenthesis open parenthesis 8 plus 2 i close parenthesis .")[$( 8 − 2 i ) ( 8 + 2 i ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[Mathematical expressions illustrating the difference of squares formula, (a - b)(a + b), and its application with complex numbers, (8 - 2i)(8 + 2i).], alt: "Mathematical expressions illustrating the difference of squares formula, (a - b)(a + b), and its application with complex numbers, (8 - 2i)(8 + 2i).", caption: none)]),
  [Use the Product of Complex Conjugates Pattern, #linebreak() #math.equation(block: false, alt: "open parenthesis a minus b i close parenthesis open parenthesis a plus b i close parenthesis equals a squared plus b squared .")[$( a − b i ) ( a + b i ) = a^(2) + b^(2) .$]], [#figure(figph[A mathematical image displaying a general algebraic expression 'a^2 + b^2' in red text, followed by a specific numerical instance '8^2 + 2^2' in black text below it.], alt: "A mathematical image displaying a general algebraic expression 'a^2 + b^2' in red text, followed by a specific numerical instance '8^2 + 2^2' in black text below it.", caption: none)],
  [Simplify the squares.], [#figure(figph[The image displays the mathematical expression '64 + 4' in black text against a white background.], alt: "The image displays the mathematical expression '64 + 4' in black text against a white background.", caption: none)],
  [Add.], [#figure(figph[The number 68 is prominently displayed in black text against a stark white background.], alt: "The number 68 is prominently displayed in black text against a stark white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply using the Product of Complex Conjugates Pattern: #math.equation(block: false, alt: "open parenthesis 3 minus 10 i close parenthesis open parenthesis 3 plus 10 i close parenthesis .")[$( 3 − 10 i ) ( 3 + 10 i ) .$]

#solutionbox[
109
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply using the Product of Complex Conjugates Pattern: #math.equation(block: false, alt: "open parenthesis −5 plus 4 i close parenthesis open parenthesis −5 minus 4 i close parenthesis .")[$( −5 + 4 i ) ( −5 − 4 i ) .$]

#solutionbox[
41
]
]


=== Divide Complex Numbers

Dividing complex numbers is much like rationalizing a denominator. We want our result to be in standard form with no imaginary numbers in the denominator.

#examplebox("Example 11")[How to Divide Complex Numbers][
Divide: #math.equation(block: false, alt: "the fraction 4 plus 3 i over 3 minus 4 i .")[$frac(4 + 3 i, 3 − 4 i) .$]

#solutionbox[
#figure(figph[Step 1 is to write both the numerator and denominator in standard form. For this example they are both in standard form.], alt: "Step 1 is to write both the numerator and denominator in standard form. For this example they are both in standard form.", caption: none)

#figure(figph[Step 2 is to multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of 3 minus 4 i is 3 plus 4 i. The resulting expression is the quantity 4 plus 3 i in parentheses times the quantity 3 plus 4 i in parentheses divided by the product of 3 minus 4 i in parentheses and the quantity 3 plus 4 i in parentheses.], alt: "Step 2 is to multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of 3 minus 4 i is 3 plus 4 i. The resulting expression is the quantity 4 plus 3 i in parentheses times the quantity 3 plus 4 i in parentheses divided by the product of 3 minus 4 i in parentheses and the quantity 3 plus 4 i in parentheses.", caption: none)

#figure(figph[Step 3 is to simplify and write the result in standard form. Use the pattern the quantity a plus b i in parentheses equals a squared plus b squared in the denominator. The expression for this example then becomes the quantity 12 plus 16 i plus 9 i plus 12 i squared in parentheses divided by the sum of 9 and 16. Combining like terms we get the quantity 12 plus 25 i minus 12 in parentheses divided by 25. Simplifying we get 25 i divided by 25. Write the result in standard form. The result is i.], alt: "Step 3 is to simplify and write the result in standard form. Use the pattern the quantity a plus b i in parentheses equals a squared plus b squared in the denominator. The expression for this example then becomes the quantity 12 plus 16 i plus 9 i plus 12 i squared in parentheses divided by the sum of 9 and 16. Combining like terms we get the quantity 12 plus 25 i minus 12 in parentheses divided by 25. Simplifying we get 25 i divided by 25. Write the result in standard form. The result is i.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction 2 plus 5 i over 5 minus 2 i .")[$frac(2 + 5 i, 5 − 2 i) .$]

#solutionbox[
#emph[i]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction 1 plus 6 i over 6 minus i .")[$frac(1 + 6 i, 6 − i) .$]

#solutionbox[
#emph[i]
]
]

We summarize the steps here.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[How to divide complex numbers.]

+ Write both the numerator and denominator in standard form.
+ Multiply the numerator and denominator by the complex conjugate of the denominator.
+ Simplify and write the result in standard form.
]

#examplebox("Example 12")[][
Divide, writing the answer in standard form: #math.equation(block: false, alt: "the fraction −3 over 5 plus 2 i .")[$frac(−3, 5 + 2 i) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction −3 over 5 plus 2 i")[$frac(−3, 5 + 2 i)$]]),
  [Multiply the numerator and denominator by the #linebreak() complex conjugate of the denominator.], [#math.equation(block: false, alt: "the fraction −3 open parenthesis 5 minus 2 i close parenthesis over open parenthesis 5 plus 2 i close parenthesis open parenthesis 5 minus 2 i close parenthesis")[$frac(−3 ( 5 − 2 i ), ( 5 + 2 i ) ( 5 − 2 i ))$]],
  [Multiply in the numerator and use the Product of #linebreak() Complex Conjugates Pattern in the denominator.], [#math.equation(block: false, alt: "the fraction −15 plus 6 i over 5 squared plus 2 squared")[$frac(−15 + 6 i, 5^(2) + 2^(2))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction −15 plus 6 i over 29")[$frac(−15 + 6 i, 29)$]],
  [Write in standard form.], [#math.equation(block: false, alt: "minus the fraction 15 over 29 plus the fraction 6 over 29 i")[$− frac(15, 29) + frac(6, 29) i$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide, writing the answer in standard form: #math.equation(block: false, alt: "the fraction 4 over 1 minus 4 i .")[$frac(4, 1 − 4 i) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 over 17 plus the fraction 16 over 17 i")[$frac(4, 17) + frac(16, 17) i$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide, writing the answer in standard form: #math.equation(block: false, alt: "the fraction −2 over −1 plus 2 i .")[$frac(−2, −1 + 2 i) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 over 5 plus the fraction 4 over 5 i")[$frac(2, 5) + frac(4, 5) i$]
]
]

Be careful as you find the conjugate of the denominator.

#examplebox("Example 13")[][
Divide: #math.equation(block: false, alt: "the fraction 5 plus 3 i over 4 i .")[$frac(5 + 3 i, 4 i) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 5 plus 3 i over 4 i")[$frac(5 + 3 i, 4 i)$]]),
  [Write the denominator in standard form.], [#math.equation(block: false, alt: "the fraction 5 plus 3 i over 0 plus 4 i")[$frac(5 + 3 i, 0 + 4 i)$]],
  [Multiply the numerator and denominator by #linebreak() the complex conjugate of the denominator.], [#math.equation(block: false, alt: "the fraction open parenthesis 5 plus 3 i close parenthesis open parenthesis 0 minus 4 i close parenthesis over open parenthesis 0 plus 4 i close parenthesis open parenthesis 0 minus 4 i close parenthesis")[$frac(( 5 + 3 i ) ( 0 − 4 i ), ( 0 + 4 i ) ( 0 − 4 i ))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction open parenthesis 5 plus 3 i close parenthesis open parenthesis −4 i close parenthesis over open parenthesis 4 i close parenthesis open parenthesis −4 i close parenthesis")[$frac(( 5 + 3 i ) ( −4 i ), ( 4 i ) ( −4 i ))$]],
  [Multiply.], [#math.equation(block: false, alt: "the fraction −20 i minus 12 i squared over −16 i squared")[$frac(−20 i − 12 i^(2), −16 i^(2))$]],
  [Simplify the #math.equation(block: false, alt: "i squared .")[$i^(2) .$]], [#math.equation(block: false, alt: "the fraction −20 i plus 12 over 16")[$frac(−20 i + 12, 16)$]],
  [Rewrite in standard form.], [#math.equation(block: false, alt: "the fraction 12 over 16 minus the fraction 20 over 16 i")[$#h(0.5em) frac(12, 16) − frac(20, 16) i$]],
  [Simplify the fractions.], [#math.equation(block: false, alt: "the fraction 3 over 4 minus the fraction 5 over 4 i")[$#h(0.5em) frac(3, 4) − frac(5, 4) i$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction 3 plus 3 i over 2 i .")[$frac(3 + 3 i, 2 i) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 over 2 minus the fraction 3 over 2 i")[$frac(3, 2) − frac(3, 2) i$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "the fraction 2 plus 4 i over 5 i .")[$frac(2 + 4 i, 5 i) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 over 5 minus the fraction 2 over 5 i")[$frac(4, 5) − frac(2, 5) i$]
]
]


=== Simplify Powers of #emph[i]

The powers of #math.equation(block: false, alt: "i")[$i$] make an interesting pattern that will help us simplify higher powers of #emph[i]. Let’s evaluate the powers of #math.equation(block: false, alt: "i")[$i$] to see the pattern.

#math.equation(block: true, alt: "i to the power 1, i squared, i cubed, i to the power 4; i, minus 1, i squared times i, i squared times i squared; minus 1 times i, open parenthesis −1 close parenthesis open parenthesis −1 close parenthesis; minus i, 1; i to the power 5, i to the power 6, i to the power 7, i to the power 8; i to the power 4 times i, i to the power 4 times i squared, i to the power 4 times i cubed, i to the power 4 times i to the power 4; 1 times i, 1 times i squared, 1 times i cubed, 1 times 1; i, i squared, i cubed, 1; minus 1, minus i")[$i^(1) &  &  & #h(2em) i^(2) &  &  & #h(2em) i^(3) &  &  & #h(2em) i^(4) \ i &  &  & #h(2em) − 1 &  &  & #h(2em) i^(2) · i &  &  & #h(2em) i^(2) · i^(2) \  &  &  &  &  &  & #h(2em) − 1 · i &  &  & #h(2em) ( −1 ) ( −1 ) \  &  &  &  &  &  & #h(2em) − i &  &  & #h(2em) 1 \  \  \ i^(5) &  &  & #h(2em) i^(6) &  &  & #h(2em) i^(7) &  &  & #h(2em) i^(8) \ i^(4) · i &  &  & #h(2em) i^(4) · i^(2) &  &  & #h(2em) i^(4) · i^(3) &  &  & #h(2em) i^(4) · i^(4) \ 1 · i &  &  & #h(2em) 1 · i^(2) &  &  & #h(2em) 1 · i^(3) &  &  & #h(2em) 1 · 1 \ i &  &  & #h(2em) i^(2) &  &  & #h(2em) i^(3) &  &  & #h(2em) 1 \  &  &  & #h(2em) − 1 &  &  & #h(2em) − i$]We summarize this now.

#math.equation(block: true, alt: "i to the power 1, equals, i, i to the power 5, equals, i; i squared, equals, −1, i to the power 6, equals, −1; i cubed, equals, − i, i to the power 7, equals, − i; i to the power 4, equals, 1, i to the power 8, equals, 1")[$i^(1) & = & i &  &  & #h(3em) i^(5) & = & i \ i^(2) & = & −1 &  &  & #h(3em) i^(6) & = & −1 \ i^(3) & = & "−" i &  &  & #h(3em) i^(7) & = & "−" i \ i^(4) & = & 1 &  &  & #h(3em) i^(8) & = & 1$]If we continued, the pattern would keep repeating in blocks of four. We can use this pattern to help us simplify powers of #emph[i]. Since #emph[i]#super[4] = 1, we rewrite each power, #emph[i#super[n]], as a product using #emph[i]#super[4] to a power and another power of #emph[i].

We rewrite it in the form #math.equation(block: false, alt: "i to the power n equals open parenthesis i to the power 4 close parenthesis to the power q times i to the power r ,")[$i^(n) = attach(( i^(4) ), t: q) · i^(r) ,$] where the exponent, #emph[q], is the quotient of #emph[n] divided by 4 and the exponent, #emph[r], is the remainder from this division. For example, to simplify #emph[i]#super[57], we divide 57 by 4 and we get 14 with a remainder of 1. In other words, #math.equation(block: false, alt: "57 equals 4 times 14 plus 1 .")[$57 = 4 · 14 + 1 .$] So we write #math.equation(block: false, alt: "i to the power 57 equals open parenthesis 1 to the power 4 close parenthesis to the power 14 times i to the power 1")[$i^(57) = attach(( 1^(4) ), t: 14) · i^(1)$] and then simplify from there.

#figure(figph[A mathematical problem demonstrating the simplification of i to the power of 57, using long division (57 / 4 = 14 remainder 1) to determine that i^57 equals i.], alt: "A mathematical problem demonstrating the simplification of i to the power of 57, using long division (57 / 4 = 14 remainder 1) to determine that i^57 equals i.", caption: none)

#examplebox("Example 14")[][
Simplify: #math.equation(block: false, alt: "i to the power 86 .")[$i^(86) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "i to the power 86")[$i^(86)$]]),
  [Divide 86 by 4 and rewrite #math.equation(block: false, alt: "i to the power 86")[$i^(86)$] in the #linebreak() #math.equation(block: false, alt: "i to the power n equals open parenthesis i to the power 4 close parenthesis to the power q times i to the power r")[$i^(n) = attach(( i^(4) ), t: q) · i^(r)$] form.], [#math.equation(block: false, alt: "open parenthesis 1 to the power 4 close parenthesis to the power 21 times i squared")[$attach(( 1^(4) ), t: 21) · i^(2)$]],
  [], [#figure(figph[A long division calculation of 86 divided by 4. The steps show that 4 goes into 8 two times, resulting in 8. Bringing down the 6, 4 goes into 6 one time, resulting in 4, with a remainder of 2. The quotient is 21 with a remainder of 2.], alt: "A long division calculation of 86 divided by 4. The steps show that 4 goes into 8 two times, resulting in 8. Bringing down the 6, 4 goes into 6 one time, resulting in 4, with a remainder of 2. The quotient is 21 with a remainder of 2.", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "open parenthesis 1 close parenthesis to the power 21 times open parenthesis −1 close parenthesis")[$attach(( 1 ), t: 21) · ( −1 )$]],
  [Simplify.], [#math.equation(block: false, alt: "–1")[$–1$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "i to the power 75 .")[$i^(75) .$]

#solutionbox[
#math.equation(block: true, alt: "− i")[$"−" i$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "i to the power 92 .")[$i^(92) .$]

#solutionbox[
#math.equation(block: true, alt: "1")[$1$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Access these online resources for additional instruction and practice with the complex number system.

- #link("https://openstax.org/l/37CompNumb1")[Expressing Square Roots of Negative Numbers with i]
- #link("https://openstax.org/l/37CompNumb2")[Subtract and Multiply Complex Numbers]
- #link("https://openstax.org/l/37CompNumb3")[Dividing Complex Numbers]
- #link("https://openstax.org/l/37CompNumb4")[Rewriting Powers of i]
]


=== Key Concepts

- #strong[Square Root of a Negative Number]
- If #emph[b] is a positive real number, then #math.equation(block: false, alt: "the square root of − b equals the square root of b i")[$sqrt("−" b) = sqrt(b) i$] #linebreak() 
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "a plus b i")[$a + b i$]], []),
  [#math.equation(block: false, alt: "b equals 0")[$b = 0$]], [#math.equation(block: false, alt: "a plus 0 times i; a")[$a + 0 · i \  \ #h(1.2em) a$]], [Real number],
  [#math.equation(block: false, alt: "b not equal to 0")[$b ≠ 0$]], [#math.equation(block: false, alt: "a plus b i")[$a + b i$]], [Imaginary number],
  [#math.equation(block: false, alt: "a equals 0")[$a = 0$]], [#math.equation(block: false, alt: "0 plus b i; b i")[$0 + b i \  \  \ #h(0.8em) b i$]], [Pure imaginary number],
))
- A complex number is in #strong[standard form] when written as #emph[a] + #emph[bi], where #emph[a, b] are real numbers. #linebreak() 
#figure(figph[The diagram has a rectangle with the labels “Complex Numbers” and a plus b i. A second rectangle has the labels “Real Numbers”, a plus b i, b = 0. A third rectangle has the labels “Imaginary Numbers”, a plus b i, b not equal to 0. Arrows go from the Real Numbers rectangle and Imaginary Numbers rectangle and point toward the Complex Numbers rectangle.], alt: "The diagram has a rectangle with the labels “Complex Numbers” and a plus b i. A second rectangle has the labels “Real Numbers”, a plus b i, b = 0. A third rectangle has the labels “Imaginary Numbers”, a plus b i, b not equal to 0. Arrows go from the Real Numbers rectangle and Imaginary Numbers rectangle and point toward the Complex Numbers rectangle.", caption: none)
- #strong[Product of Complex Conjugates]
- If #emph[a, b] are real numbers, then #linebreak() 
#math.equation(block: false, alt: "open parenthesis a minus b i close parenthesis open parenthesis a plus b i close parenthesis equals a squared plus b squared")[$( a − b i ) ( a + b i ) = a^(2) + b^(2)$]
- #strong[How to Divide Complex Numbers]
+ Write both the numerator and denominator in standard form.
+ Multiply the numerator and denominator by the complex conjugate of the denominator.
+ Simplify and write the result in standard form.


=== Section Exercises


==== Practice Makes Perfect

#strong[Evaluate the Square Root of a Negative Number]

In the following exercises, write each expression in terms of #emph[i] and simplify if possible.

ⓐ #math.equation(block: false, alt: "the square root of −16")[$sqrt(−16)$] ⓑ #math.equation(block: false, alt: "the square root of −11")[$sqrt(−11)$] #linebreak() ⓒ #math.equation(block: false, alt: "the square root of −8")[$sqrt(−8)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "4 i")[$4 i$] ⓑ #math.equation(block: false, alt: "the square root of 11 i")[$sqrt(11) i$] ⓒ #math.equation(block: false, alt: "2 the square root of 2 i")[$2 sqrt(2) i$]
]

ⓐ #math.equation(block: false, alt: "the square root of −121")[$sqrt(−121)$] ⓑ #math.equation(block: false, alt: "the square root of −1")[$sqrt(−1)$] ⓒ #math.equation(block: false, alt: "the square root of −20")[$sqrt(−20)$]

ⓐ #math.equation(block: false, alt: "the square root of −100")[$sqrt(−100)$] ⓑ #math.equation(block: false, alt: "the square root of −13")[$sqrt(−13)$] ⓒ #math.equation(block: false, alt: "the square root of −45")[$sqrt(−45)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "10 i")[$10 i$] ⓑ #math.equation(block: false, alt: "the square root of 13 i")[$sqrt(13) i$] ⓒ #math.equation(block: false, alt: "3 the square root of 5 i")[$3 sqrt(5) i$]
]

ⓐ #math.equation(block: false, alt: "the square root of −49")[$sqrt(−49)$] ⓑ #math.equation(block: false, alt: "the square root of −15")[$sqrt(−15)$] ⓒ #math.equation(block: false, alt: "the square root of −75")[$sqrt(−75)$]

#strong[Add or Subtract Complex Numbers] In the following exercises, add or subtract.

#math.equation(block: true, alt: "the square root of −75 plus the square root of −48")[$sqrt(−75) + sqrt(−48)$]

#solutionbox[
#math.equation(block: true, alt: "9 the square root of 3 i")[$9 sqrt(3) i$]
]

#math.equation(block: true, alt: "the square root of −12 plus the square root of −75")[$sqrt(−12) + sqrt(−75)$]

#math.equation(block: true, alt: "the square root of −50 plus the square root of −18")[$sqrt(−50) + sqrt(−18)$]

#solutionbox[
#math.equation(block: true, alt: "8 the square root of 2 i")[$8 sqrt(2) i$]
]

#math.equation(block: true, alt: "the square root of −72 plus the square root of −8")[$sqrt(−72) + sqrt(−8)$]

#math.equation(block: true, alt: "open parenthesis 1 plus 3 i close parenthesis plus open parenthesis 7 plus 4 i close parenthesis")[$( 1 + 3 i ) + ( 7 + 4 i )$]

#solutionbox[
#math.equation(block: true, alt: "8 plus 7 i")[$8 + 7 i$]
]

#math.equation(block: true, alt: "open parenthesis 6 plus 2 i close parenthesis plus open parenthesis 3 minus 4 i close parenthesis")[$( 6 + 2 i ) + ( 3 − 4 i )$]

#math.equation(block: true, alt: "open parenthesis 8 minus i close parenthesis plus open parenthesis 6 plus 3 i close parenthesis")[$( 8 − i ) + ( 6 + 3 i )$]

#solutionbox[
#math.equation(block: true, alt: "14 plus 2 i")[$14 + 2 i$]
]

#math.equation(block: true, alt: "open parenthesis 7 minus 4 i close parenthesis plus open parenthesis −2 minus 6 i close parenthesis")[$( 7 − 4 i ) + ( −2 − 6 i )$]

#math.equation(block: true, alt: "open parenthesis 1 minus 4 i close parenthesis minus open parenthesis 3 minus 6 i close parenthesis")[$( 1 − 4 i ) − ( 3 − 6 i )$]

#solutionbox[
#math.equation(block: true, alt: "−2 plus 2 i")[$−2 + 2 i$]
]

#math.equation(block: true, alt: "open parenthesis 8 minus 4 i close parenthesis minus open parenthesis 3 plus 7 i close parenthesis")[$( 8 − 4 i ) − ( 3 + 7 i )$]

#math.equation(block: true, alt: "open parenthesis 6 plus i close parenthesis minus open parenthesis −2 minus 4 i close parenthesis")[$( 6 + i ) − ( −2 − 4 i )$]

#solutionbox[
#math.equation(block: true, alt: "8 plus 5 i")[$8 + 5 i$]
]

#math.equation(block: true, alt: "open parenthesis −2 plus 5 i close parenthesis minus open parenthesis −5 plus 6 i close parenthesis")[$( −2 + 5 i ) − ( −5 + 6 i )$]

#math.equation(block: true, alt: "open parenthesis 5 minus the square root of −36 close parenthesis plus open parenthesis 2 minus the square root of −49 close parenthesis")[$( 5 − sqrt(−36) ) + ( 2 − sqrt(−49) )$]

#solutionbox[
#math.equation(block: true, alt: "7 minus 13 i")[$7 − 13 i$]
]

#math.equation(block: true, alt: "open parenthesis −3 plus the square root of −64 close parenthesis plus open parenthesis 5 minus the square root of −16 close parenthesis")[$( −3 + sqrt(−64) ) + ( 5 − sqrt(−16) )$]

#math.equation(block: true, alt: "open parenthesis −7 minus the square root of −50 close parenthesis minus open parenthesis −32 minus the square root of −18 close parenthesis")[$( −7 − sqrt(−50) ) − ( −32 − sqrt(−18) )$]

#solutionbox[
#math.equation(block: true, alt: "25 minus 2 the square root of 2 i")[$25 − 2 sqrt(2) i$]
]

#math.equation(block: true, alt: "open parenthesis −5 plus the square root of −27 close parenthesis minus open parenthesis −4 minus the square root of −48 close parenthesis")[$( −5 + sqrt(−27) ) − ( −4 − sqrt(−48) )$]

#strong[Multiply Complex Numbers]

In the following exercises, multiply.

#math.equation(block: true, alt: "4 i open parenthesis 5 minus 3 i close parenthesis")[$4 i ( 5 − 3 i )$]

#solutionbox[
#math.equation(block: true, alt: "12 plus 20 i")[$12 + 20 i$]
]

#math.equation(block: true, alt: "2 i open parenthesis −3 plus 4 i close parenthesis")[$2 i ( −3 + 4 i )$]

#math.equation(block: true, alt: "−6 i open parenthesis −3 minus 2 i close parenthesis")[$−6 i ( −3 − 2 i )$]

#solutionbox[
#math.equation(block: true, alt: "−12 plus 18 i")[$−12 + 18 i$]
]

#math.equation(block: true, alt: "− i open parenthesis 6 plus 5 i close parenthesis")[$"−" i ( 6 + 5 i )$]

#math.equation(block: true, alt: "open parenthesis 4 plus 3 i close parenthesis open parenthesis −5 plus 6 i close parenthesis")[$( 4 + 3 i ) ( −5 + 6 i )$]

#solutionbox[
#math.equation(block: true, alt: "−38 plus plus 9 i")[$−38 + + 9 i$]
]

#math.equation(block: true, alt: "open parenthesis −2 minus 5 i close parenthesis open parenthesis −4 plus 3 i close parenthesis")[$( −2 − 5 i ) ( −4 + 3 i )$]

#math.equation(block: true, alt: "open parenthesis −3 plus 3 i close parenthesis open parenthesis −2 minus 7 i close parenthesis")[$( −3 + 3 i ) ( −2 − 7 i )$]

#solutionbox[
#math.equation(block: true, alt: "27 plus 15 i")[$27 + 15 i$]
]

#math.equation(block: true, alt: "open parenthesis −6 minus 2 i close parenthesis open parenthesis −3 minus 5 i close parenthesis")[$( −6 − 2 i ) ( −3 − 5 i )$]

In the following exercises, multiply using the Product of Binomial Squares Pattern.

#math.equation(block: true, alt: "open parenthesis 3 plus 4 i close parenthesis squared")[$attach(( 3 + 4 i ), t: 2)$]

#solutionbox[
#math.equation(block: true, alt: "−7 plus 24 i")[$−7 + 24 i$]
]

#math.equation(block: true, alt: "open parenthesis −1 plus 5 i close parenthesis squared")[$attach(( −1 + 5 i ), t: 2)$]

#math.equation(block: true, alt: "open parenthesis −2 minus 3 i close parenthesis squared")[$attach(( −2 − 3 i ), t: 2)$]

#solutionbox[
#math.equation(block: true, alt: "−5 plus 12 i")[$−5 + 12 i$]
]

#math.equation(block: true, alt: "open parenthesis −6 minus 5 i close parenthesis squared")[$attach(( −6 − 5 i ), t: 2)$]

In the following exercises, multiply.

#math.equation(block: true, alt: "the square root of −25 times the square root of −36")[$sqrt(−25) · sqrt(−36)$]

#solutionbox[
#math.equation(block: true, alt: "−30")[$−30$]
]

#math.equation(block: true, alt: "the square root of −4 times the square root of −16")[$sqrt(−4) · sqrt(−16)$]

#math.equation(block: true, alt: "the square root of −9 times the square root of −100")[$sqrt(−9) · sqrt(−100)$]

#solutionbox[
#math.equation(block: true, alt: "−30")[$−30$]
]

#math.equation(block: true, alt: "the square root of −64 times the square root of −9")[$sqrt(−64) · sqrt(−9)$]

#math.equation(block: true, alt: "open parenthesis −2 minus the square root of −27 close parenthesis open parenthesis 4 minus the square root of −48 close parenthesis")[$( −2 − sqrt(−27) ) ( 4 − sqrt(−48) )$]

#solutionbox[
#math.equation(block: true, alt: "−44 minus 4 i the square root of 3")[$−44 − 4 i sqrt(3)$]
]

#math.equation(block: true, alt: "open parenthesis 5 minus the square root of −12 close parenthesis open parenthesis −3 plus the square root of −75 close parenthesis")[$( 5 − sqrt(−12) ) ( −3 + sqrt(−75) )$]

#math.equation(block: true, alt: "open parenthesis 2 plus the square root of −8 close parenthesis open parenthesis −4 plus the square root of −18 close parenthesis")[$( 2 + sqrt(−8) ) ( −4 + sqrt(−18) )$]

#solutionbox[
#math.equation(block: true, alt: "−20 minus 2 the square root of 2 i")[$−20 − 2 sqrt(2) i$]
]

#math.equation(block: true, alt: "open parenthesis 5 plus the square root of −18 close parenthesis open parenthesis −2 minus the square root of −50 close parenthesis")[$( 5 + sqrt(−18) ) ( −2 − sqrt(−50) )$]

#math.equation(block: true, alt: "open parenthesis 2 minus i close parenthesis open parenthesis 2 plus i close parenthesis")[$( 2 − i ) ( 2 + i )$]

#solutionbox[
5
]

#math.equation(block: true, alt: "open parenthesis 4 minus 5 i close parenthesis open parenthesis 4 plus 5 i close parenthesis")[$( 4 − 5 i ) ( 4 + 5 i )$]

#math.equation(block: true, alt: "open parenthesis 7 minus 2 i close parenthesis open parenthesis 7 plus 2 i close parenthesis")[$( 7 − 2 i ) ( 7 + 2 i )$]

#solutionbox[
53
]

#math.equation(block: true, alt: "open parenthesis −3 minus 8 i close parenthesis open parenthesis −3 plus 8 i close parenthesis")[$( −3 − 8 i ) ( −3 + 8 i )$]

In the following exercises, multiply using the Product of Complex Conjugates Pattern.

#math.equation(block: true, alt: "open parenthesis 7 minus i close parenthesis open parenthesis 7 plus i close parenthesis")[$( 7 − i ) ( 7 + i )$]

#solutionbox[
50
]

#math.equation(block: true, alt: "open parenthesis 6 minus 5 i close parenthesis open parenthesis 6 plus 5 i close parenthesis")[$( 6 − 5 i ) ( 6 + 5 i )$]

#math.equation(block: true, alt: "open parenthesis 9 minus 2 i close parenthesis open parenthesis 9 plus 2 i close parenthesis")[$( 9 − 2 i ) ( 9 + 2 i )$]

#solutionbox[
85
]

#math.equation(block: true, alt: "open parenthesis −3 minus 4 i close parenthesis open parenthesis −3 plus 4 i close parenthesis")[$( −3 − 4 i ) ( −3 + 4 i )$]

#strong[Divide Complex Numbers]

In the following exercises, divide.

#math.equation(block: true, alt: "the fraction 3 plus 4 i over 4 minus 3 i")[$frac(3 + 4 i, 4 − 3 i)$]

#solutionbox[
#emph[i]
]

#math.equation(block: true, alt: "the fraction 5 minus 2 i over 2 plus 5 i")[$frac(5 − 2 i, 2 + 5 i)$]

#math.equation(block: true, alt: "the fraction 2 plus i over 3 minus 4 i")[$frac(2 + i, 3 − 4 i)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 over 25 plus the fraction 11 over 25 i")[$frac(2, 25) + frac(11, 25) i$]
]

#math.equation(block: true, alt: "the fraction 3 minus 2 i over 6 plus i")[$frac(3 − 2 i, 6 + i)$]

#math.equation(block: true, alt: "the fraction 3 over 2 minus 3 i")[$frac(3, 2 − 3 i)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 6 over 13 plus the fraction 9 over 13 i")[$frac(6, 13) + frac(9, 13) i$]
]

#math.equation(block: true, alt: "the fraction 2 over 4 minus 5 i")[$frac(2, 4 − 5 i)$]

#math.equation(block: true, alt: "the fraction −4 over 3 minus 2 i")[$frac(−4, 3 − 2 i)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 12 over 13 minus the fraction 8 over 13 i")[$− frac(12, 13) − frac(8, 13) i$]
]

#math.equation(block: true, alt: "the fraction −1 over 3 plus 2 i")[$frac(−1, 3 + 2 i)$]

#math.equation(block: true, alt: "the fraction 1 plus 4 i over 3 i")[$frac(1 + 4 i, 3 i)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 over 3 minus the fraction 1 over 3 i")[$frac(4, 3) − frac(1, 3) i$]
]

#math.equation(block: true, alt: "the fraction 4 plus 3 i over 7 i")[$frac(4 + 3 i, 7 i)$]

#math.equation(block: true, alt: "the fraction −2 minus 3 i over 4 i")[$frac(−2 − 3 i, 4 i)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 3 over 4 plus the fraction 1 over 2 i")[$− frac(3, 4) + frac(1, 2) i$]
]

#math.equation(block: true, alt: "the fraction −3 minus 5 i over 2 i")[$frac(−3 − 5 i, 2 i)$]

#strong[Simplify Powers of #emph[i]]

In the following exercises, simplify.

#math.equation(block: true, alt: "i to the power 41")[$i^(41)$]

#solutionbox[
#emph[i]
]

#math.equation(block: true, alt: "i to the power 39")[$i^(39)$]

#math.equation(block: true, alt: "i to the power 66")[$i^(66)$]

#solutionbox[
#math.equation(block: true, alt: "−1")[$−1$]
]

#math.equation(block: true, alt: "i to the power 48")[$i^(48)$]

#math.equation(block: true, alt: "i to the power 128")[$i^(128)$]

#solutionbox[
1
]

#math.equation(block: true, alt: "i to the power 162")[$i^(162)$]

#math.equation(block: true, alt: "i to the power 137")[$i^(137)$]

#solutionbox[
#emph[i]
]

#math.equation(block: true, alt: "i to the power 255")[$i^(255)$]


==== Writing Exercises

Explain the relationship between real numbers and complex numbers.

#solutionbox[
Answers will vary.
]

Aniket multiplied as follows and he got the wrong answer. What is wrong with his reasoning?

#math.equation(block: true, alt: "the square root of −7 times the square root of −7; the square root of 49; 7")[$sqrt(−7) · sqrt(−7) \ sqrt(49) \ 7$]

Why is #math.equation(block: false, alt: "the square root of −64 equals 8 i")[$sqrt(−64) = 8 i$] but #math.equation(block: false, alt: "the cube root of −64 equals −4 .")[$root(3, −64) = −4 .$]

#solutionbox[
Answers will vary.
]

Explain how dividing complex numbers is similar to rationalizing a denominator.


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[The table has 4 columns and 4 rows. The first row is a header row with the headers “I can…”, “Confidently”, “With some help.”, and “No – I don’t get it!”. The first column contains the phrases “evaluate the square root of a negative number”, “add or subtract complex numbers”, “multiply complex numbers”, “divide complex numbers”, and “simplify powers of i”. The other columns are left blank so the learner can indicate their level of understanding.], alt: "The table has 4 columns and 4 rows. The first row is a header row with the headers “I can…”, “Confidently”, “With some help.”, and “No – I don’t get it!”. The first column contains the phrases “evaluate the square root of a negative number”, “add or subtract complex numbers”, “multiply complex numbers”, “divide complex numbers”, and “simplify powers of i”. The other columns are left blank so the learner can indicate their level of understanding.", caption: none)

ⓑ On a scale of #math.equation(block: false, alt: "1 minus 10 ,")[$1 − 10 ,$] how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?


=== Chapter Review Exercises


==== Simplify Expressions with Roots

#strong[Simplify Expressions with Roots]

In the following exercises, simplify.

ⓐ #math.equation(block: false, alt: "the square root of 225")[$sqrt(225)$] ⓑ #math.equation(block: false, alt: "− the square root of 16")[$"−" sqrt(16)$]

#solutionbox[
ⓐ 15 ⓑ #math.equation(block: false, alt: "−4")[$−4$]
]

ⓐ #math.equation(block: false, alt: "− the square root of 169")[$"−" sqrt(169)$] ⓑ #math.equation(block: false, alt: "the square root of −8")[$sqrt(−8)$]

ⓐ #math.equation(block: false, alt: "the cube root of 8")[$root(3, 8)$] ⓑ #math.equation(block: false, alt: "the 4th root of 81")[$root(4, 81)$] ⓒ #math.equation(block: false, alt: "the 5th root of 243")[$root(5, 243)$]

#solutionbox[
ⓐ 2 ⓑ 3 ⓒ 3
]

ⓐ #math.equation(block: false, alt: "the cube root of −512")[$root(3, −512)$] ⓑ #math.equation(block: false, alt: "the 4th root of −81")[$root(4, −81)$] ⓒ #math.equation(block: false, alt: "the 5th root of −1")[$root(5, −1)$]

#strong[Estimate and Approximate Roots]

In the following exercises, estimate each root between two consecutive whole numbers.

ⓐ #math.equation(block: false, alt: "the square root of 68")[$sqrt(68)$] ⓑ #math.equation(block: false, alt: "the cube root of 84")[$root(3, 84)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "8 less than the square root of 68 less than 9")[$8 < sqrt(68) < 9$] #linebreak() ⓑ #math.equation(block: false, alt: "4 less than the cube root of 84 less than 5")[$4 < root(3, 84) < 5$]
]

In the following exercises, approximate each root and round to two decimal places.

ⓐ #math.equation(block: false, alt: "the square root of 37")[$sqrt(37)$] ⓑ #math.equation(block: false, alt: "the cube root of 84")[$root(3, 84)$] ⓒ #math.equation(block: false, alt: "the 4th root of 125")[$root(4, 125)$]

#strong[Simplify Variable Expressions with Roots]

In the following exercises, simplify using absolute values as necessary.

ⓐ #math.equation(block: false, alt: "the cube root of a cubed")[$root(3, a^(3))$] #linebreak() ⓑ #math.equation(block: false, alt: "the 7th root of b to the power 7")[$root(7, b^(7))$]

#solutionbox[
ⓐ #emph[a] ⓑ #math.equation(block: false, alt: "b")[$b$]
]

ⓐ #math.equation(block: false, alt: "the square root of a to the power 14")[$sqrt(a^(14))$] #linebreak() ⓑ #math.equation(block: false, alt: "the square root of w to the power 24")[$sqrt(w^(24))$]

ⓐ #math.equation(block: false, alt: "the 4th root of m to the power 8")[$root(4, m^(8))$] #linebreak() ⓑ #math.equation(block: false, alt: "the 5th root of n to the power 20")[$root(5, n^(20))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "m squared")[$m^(2)$] ⓑ #math.equation(block: false, alt: "n to the power 4")[$n^(4)$]
]

ⓐ #math.equation(block: false, alt: "the square root of 121 m to the power 20")[$sqrt(121 m^(20))$] #linebreak() ⓑ #math.equation(block: false, alt: "− the square root of 64 a squared")[$"−" sqrt(64 a^(2))$]

ⓐ #math.equation(block: false, alt: "the cube root of 216 a to the power 6")[$root(3, 216 a^(6))$] #linebreak() ⓑ #math.equation(block: false, alt: "the 5th root of 32 b to the power 20")[$root(5, 32 b^(20))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "6 a squared")[$6 a^(2)$] ⓑ #math.equation(block: false, alt: "2 b to the power 4")[$2 b^(4)$]
]

ⓐ #math.equation(block: false, alt: "the square root of 144 x squared y squared")[$sqrt(144 x^(2) y^(2))$] #linebreak() ⓑ #math.equation(block: false, alt: "the square root of 169 w to the power 8 y to the power 10")[$sqrt(169 w^(8) y^(10))$] #linebreak() ⓒ #math.equation(block: false, alt: "the cube root of 8 a to the power 51 b to the power 6")[$root(3, 8 a^(51) b^(6))$]


==== Simplify Radical Expressions

#strong[Use the Product Property to Simplify Radical Expressions]

In the following exercises, use the Product Property to simplify radical expressions.

#math.equation(block: true, alt: "the square root of 125")[$sqrt(125)$]

#solutionbox[
#math.equation(block: true, alt: "5 the square root of 5")[$5 sqrt(5)$]
]

#math.equation(block: true, alt: "the square root of 675")[$sqrt(675)$]

ⓐ #math.equation(block: false, alt: "the cube root of 625")[$root(3, 625)$] ⓑ #math.equation(block: false, alt: "the 6th root of 128")[$root(6, 128)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "5 the cube root of 5")[$5 root(3, 5)$] ⓑ #math.equation(block: false, alt: "2 the 6th root of 2")[$2 root(6, 2)$]
]

In the following exercises, simplify using absolute value signs as needed.

ⓐ #math.equation(block: false, alt: "the square root of a to the power 23")[$sqrt(a^(23))$] #linebreak() ⓑ #math.equation(block: false, alt: "the cube root of b to the power 8")[$root(3, b^(8))$] #linebreak() ⓒ #math.equation(block: false, alt: "the 8th root of c to the power 13")[$root(8, c^(13))$]

ⓐ #math.equation(block: false, alt: "the square root of 80 s to the power 15")[$sqrt(80 s^(15))$] #linebreak() ⓑ #math.equation(block: false, alt: "the 5th root of 96 a to the power 7")[$root(5, 96 a^(7))$] #linebreak() ⓒ #math.equation(block: false, alt: "the 6th root of 128 b to the power 7")[$root(6, 128 b^(7))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "4 | s to the power 7 | the root of 5 s")[$4 | s^(7) | root(, 5 s)$] ⓑ #math.equation(block: false, alt: "2 a the 5th root of 3 a squared")[$2 a root(5, 3 a^(2))$] #linebreak() ⓒ #math.equation(block: false, alt: "2 | b | the 6th root of 2 b")[$2 | b | root(6, 2 b)$]
]

ⓐ #math.equation(block: false, alt: "the square root of 96 r cubed s cubed")[$sqrt(96 r^(3) s^(3))$] #linebreak() ⓑ #math.equation(block: false, alt: "the cube root of 80 x to the power 7 y to the power 6")[$root(3, 80 x^(7) y^(6))$] #linebreak() ⓒ #math.equation(block: false, alt: "the 4th root of 80 x to the power 8 y to the power 9")[$root(4, 80 x^(8) y^(9))$]

ⓐ #math.equation(block: false, alt: "the 5th root of −32")[$root(5, −32)$] #linebreak() ⓑ #math.equation(block: false, alt: "the 8th root of −1")[$root(8, −1)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−2")[$−2$] ⓑ not real
]

ⓐ #math.equation(block: false, alt: "8 plus the square root of 96")[$8 + sqrt(96)$] #linebreak() ⓑ #math.equation(block: false, alt: "the fraction 2 plus the square root of 40 over 2")[$frac(2 + sqrt(40), 2)$]

#strong[Use the Quotient Property to Simplify Radical Expressions]

In the following exercises, use the Quotient Property to simplify square roots.

ⓐ #math.equation(block: false, alt: "the square root of the fraction 72 over 98")[$sqrt(frac(72, 98))$] ⓑ #math.equation(block: false, alt: "the cube root of the fraction 24 over 81")[$root(3, frac(24, 81))$] ⓒ #math.equation(block: false, alt: "the 4th root of the fraction 6 over 96")[$root(4, frac(6, 96))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction 6 over 7")[$frac(6, 7)$] ⓑ #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] ⓒ #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]
]

ⓐ #math.equation(block: false, alt: "the square root of the fraction y to the power 4 over y to the power 8")[$sqrt(frac(y^(4), y^(8)))$] ⓑ #math.equation(block: false, alt: "the 5th root of the fraction u to the power 21 over u to the power 11")[$root(5, frac(u^(21), u^(11)))$] ⓒ #math.equation(block: false, alt: "the 6th root of the fraction v to the power 30 over v to the power 12")[$root(6, frac(v^(30), v^(12)))$]

#math.equation(block: true, alt: "the square root of the fraction 300 m to the power 5 over 64")[$sqrt(frac(300 m^(5), 64))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 m squared the square root of 3 m over 4")[$frac(5 m^(2) sqrt(3 m), 4)$]
]

ⓐ #math.equation(block: false, alt: "the square root of the fraction 28 p to the power 7 over q squared")[$sqrt(frac(28 p^(7), q^(2)))$] #linebreak() ⓑ #math.equation(block: false, alt: "the cube root of the fraction 81 s to the power 8 over t cubed")[$root(3, frac(81 s^(8), t^(3)))$] #linebreak() ⓒ #math.equation(block: false, alt: "the 4th root of the fraction 64 p to the power 15 over q to the power 12")[$root(4, frac(64 p^(15), q^(12)))$]

ⓐ #math.equation(block: false, alt: "the square root of the fraction 27 p squared q over 108 p to the power 4 q cubed")[$sqrt(frac(27 p^(2) q, 108 p^(4) q^(3)))$] #linebreak() ⓑ #math.equation(block: false, alt: "the cube root of the fraction 16 c to the power 5 d to the power 7 over 250 c squared d squared")[$root(3, frac(16 c^(5) d^(7), 250 c^(2) d^(2)))$] #linebreak() ⓒ #math.equation(block: false, alt: "the 6th root of the fraction 2 m to the power 9 n to the power 7 over 128 m cubed n")[$root(6, frac(2 m^(9) n^(7), 128 m^(3) n))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction 1 over 2 | p q |")[$frac(1, 2 | p q |)$] ⓑ #math.equation(block: false, alt: "the fraction 2 c d over 5 the cube root of d squared")[$frac(2 c d, 5) root(3, d^(2))$] #linebreak() ⓒ #math.equation(block: false, alt: "the fraction | m n | over 2")[$frac(| m n |, 2)$]
]

ⓐ #math.equation(block: false, alt: "the fraction the square root of 80 q to the power 5 over the square root of 5 q")[$frac(sqrt(80 q^(5)), sqrt(5 q))$] #linebreak() ⓑ #math.equation(block: false, alt: "the fraction the cube root of minus 625 over the cube root of 5")[$frac(root(3, − 625), root(3, 5))$] #linebreak() ⓒ #math.equation(block: false, alt: "the fraction the 4th root of 80 m to the power 7 over the 4th root of 5 m")[$frac(root(4, 80 m^(7)), root(4, 5 m))$]


==== Simplify Rational Exponents

#strong[Simplify expressions with #math.equation(block: false, alt: "a to the power the fraction 1 over n")[$a^(frac(1, n))$]]

In the following exercises, write as a radical expression.

ⓐ #math.equation(block: false, alt: "r to the power the fraction 1 over 2")[$r^(frac(1, 2))$] ⓑ #math.equation(block: false, alt: "s to the power the fraction 1 over 3")[$s^(frac(1, 3))$] ⓒ #math.equation(block: false, alt: "t to the power the fraction 1 over 4")[$t^(frac(1, 4))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the square root of r")[$sqrt(r)$] ⓑ #math.equation(block: false, alt: "the cube root of s")[$root(3, s)$] ⓒ #math.equation(block: false, alt: "the 4th root of t")[$root(4, t)$]
]

In the following exercises, write with a rational exponent.

ⓐ #math.equation(block: false, alt: "the square root of 21 p")[$sqrt(21 p)$] ⓑ #math.equation(block: false, alt: "the 4th root of 8 q")[$root(4, 8 q)$] ⓒ #math.equation(block: false, alt: "4 the 6th root of 36 r")[$4 root(6, 36 r)$]

In the following exercises, simplify.

ⓐ #math.equation(block: false, alt: "625 to the power the fraction 1 over 4")[$625^(frac(1, 4))$] #linebreak() ⓑ #math.equation(block: false, alt: "243 to the power the fraction 1 over 5")[$243^(frac(1, 5))$] #linebreak() ⓒ #math.equation(block: false, alt: "32 to the power the fraction 1 over 5")[$32^(frac(1, 5))$]

#solutionbox[
ⓐ 5 ⓑ 3 ⓒ 2
]

ⓐ #math.equation(block: false, alt: "open parenthesis −1,000 close parenthesis to the power the fraction 1 over 3")[$attach(( −1,000 ), t: frac(1, 3))$] #linebreak() ⓑ #math.equation(block: false, alt: "− 1,000 to the power the fraction 1 over 3")[$"−" 1,000^(frac(1, 3))$] #linebreak() ⓒ #math.equation(block: false, alt: "open parenthesis 1,000 close parenthesis to the power minus the fraction 1 over 3")[$attach(( 1,000 ), t: − frac(1, 3))$]

ⓐ #math.equation(block: false, alt: "open parenthesis −32 close parenthesis to the power the fraction 1 over 5")[$attach(( −32 ), t: frac(1, 5))$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis 243 close parenthesis to the power minus the fraction 1 over 5")[$attach(( 243 ), t: − frac(1, 5))$] #linebreak() ⓒ #math.equation(block: false, alt: "− 125 to the power the fraction 1 over 3")[$"−" 125^(frac(1, 3))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−2")[$−2$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] ⓒ #math.equation(block: false, alt: "−5")[$−5$]
]

#strong[Simplify Expressions with #math.equation(block: false, alt: "a to the power the fraction m over n")[$a^(frac(m, n))$]]

In the following exercises, write with a rational exponent.

ⓐ #math.equation(block: false, alt: "the 4th root of r to the power 7")[$root(4, r^(7))$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis the 5th root of 2 p q close parenthesis cubed")[$attach(( root(5, 2 p q) ), t: 3)$] #linebreak() ⓒ #math.equation(block: false, alt: "the 4th root of open parenthesis the fraction 12 m over 7 n close parenthesis cubed")[$root(4, attach(( frac(12 m, 7 n) ), t: 3))$]

In the following exercises, simplify.

ⓐ #math.equation(block: false, alt: "25 to the power the fraction 3 over 2")[$25^(frac(3, 2))$] #linebreak() ⓑ #math.equation(block: false, alt: "9 to the power minus the fraction 3 over 2")[$9^(− frac(3, 2))$] #linebreak() ⓒ #math.equation(block: false, alt: "open parenthesis −64 close parenthesis to the power the fraction 2 over 3")[$attach(( −64 ), t: frac(2, 3))$]

#solutionbox[
ⓐ 125 ⓑ #math.equation(block: false, alt: "the fraction 1 over 27")[$frac(1, 27)$] ⓒ 16
]

ⓐ #math.equation(block: false, alt: "− 64 to the power the fraction 3 over 2")[$"−" 64^(frac(3, 2))$] #linebreak() ⓑ #math.equation(block: false, alt: "− 64 to the power minus the fraction 3 over 2")[$"−" 64^(− frac(3, 2))$] #linebreak() ⓒ #math.equation(block: false, alt: "open parenthesis −64 close parenthesis to the power the fraction 3 over 2")[$attach(( −64 ), t: frac(3, 2))$]

#strong[Use the Laws of Exponents to Simplify Expressions with Rational Exponents]

In the following exercises, simplify.

ⓐ #math.equation(block: false, alt: "6 to the power the fraction 5 over 2 times 6 to the power the fraction 1 over 2")[$6^(frac(5, 2)) · 6^(frac(1, 2))$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis b to the power 15 close parenthesis to the power the fraction 3 over 5")[$attach(( b^(15) ), t: frac(3, 5))$] #linebreak() ⓒ #math.equation(block: false, alt: "the fraction w to the power the fraction 2 over 7 over w to the power the fraction 9 over 7")[$frac(w^(frac(2, 7)), w^(frac(9, 7)))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "6 cubed")[$6^(3)$] ⓑ #math.equation(block: false, alt: "b to the power 9")[$b^(9)$] ⓒ #math.equation(block: false, alt: "the fraction 1 over w")[$frac(1, w)$]
]

ⓐ #math.equation(block: false, alt: "the fraction a to the power the fraction 3 over 4 times a to the power minus the fraction 1 over 4 over a to the power minus the fraction 10 over 4")[$frac(a^(frac(3, 4)) · a^(− frac(1, 4)), a^(− frac(10, 4)))$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis the fraction 27 ​ b to the power the fraction 2 over 3 ​ c to the power minus the fraction 5 over 2 over b to the power minus the fraction 7 over 3 c to the power the fraction 1 over 2 close parenthesis to the power the fraction 1 over 3")[$attach(( frac(27 "​" b^(frac(2, 3)) "​" c^(− frac(5, 2)), b^(− frac(7, 3)) c^(frac(1, 2))) ), t: frac(1, 3))$]


==== Add, Subtract and Multiply Radical Expressions

#strong[Add and Subtract Radical Expressions]

In the following exercises, simplify.

ⓐ #math.equation(block: false, alt: "7 the square root of 2 minus 3 the square root of 2")[$7 sqrt(2) − 3 sqrt(2)$] #linebreak() ⓑ #math.equation(block: false, alt: "7 the cube root of p plus 2 the cube root of p")[$7 root(3, p) + 2 root(3, p)$] #linebreak() ⓒ #math.equation(block: false, alt: "5 the cube root of x minus 3 the cube root of x")[$5 root(3, x) − 3 root(3, x)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "4 the square root of 2")[$4 sqrt(2)$] ⓑ #math.equation(block: false, alt: "9 the cube root of p")[$9 root(3, p)$] ⓒ #math.equation(block: false, alt: "2 the cube root of x")[$2 root(3, x)$]
]

ⓐ #math.equation(block: false, alt: "the square root of 11 b minus 5 the square root of 11 b plus 3 the square root of 11 b")[$sqrt(11 b) − 5 sqrt(11 b) + 3 sqrt(11 b)$] #linebreak() ⓑ #math.equation(block: false, alt: "8 the 4th root of 11 c d plus 5 the 4th root of 11 c d minus 9 the 4th root of 11 c d")[$8 root(4, 11 c d) + 5 root(4, 11 c d) − 9 root(4, 11 c d)$]

ⓐ #math.equation(block: false, alt: "the square root of 48 plus the square root of 27")[$sqrt(48) + sqrt(27)$] #linebreak() ⓑ #math.equation(block: false, alt: "the cube root of 54 plus the cube root of 128")[$root(3, 54) + root(3, 128)$] #linebreak() ⓒ #math.equation(block: false, alt: "6 the 4th root of 5 minus the fraction 3 over 2 the 4th root of 80")[$6 root(4, 5) − frac(3, 2) root(4, 80)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "7 the square root of 3")[$7 sqrt(3)$] ⓑ #math.equation(block: false, alt: "7 the cube root of 2")[$7 root(3, 2)$] ⓒ #math.equation(block: false, alt: "3 the 4th root of 5")[$3 root(4, 5)$]
]

ⓐ #math.equation(block: false, alt: "the square root of 80 c to the power 7 minus the square root of 20 c to the power 7")[$sqrt(80 c^(7)) − sqrt(20 c^(7))$] #linebreak() ⓑ #math.equation(block: false, alt: "2 the 4th root of 162 r to the power 10 plus 4 the 4th root of 32 r to the power 10")[$2 root(4, 162 r^(10)) + 4 root(4, 32 r^(10))$]

#math.equation(block: true, alt: "3 the square root of 75 y squared plus 8 y the square root of 48 minus the square root of 300 y squared")[$3 sqrt(75 y^(2)) + 8 y sqrt(48) − sqrt(300 y^(2))$]

#solutionbox[
#math.equation(block: true, alt: "37 y the square root of 3")[$37 y sqrt(3)$]
]

#strong[Multiply Radical Expressions]

In the following exercises, simplify.

ⓐ #math.equation(block: false, alt: "open parenthesis 5 the square root of 6 close parenthesis open parenthesis − the square root of 12 close parenthesis")[$( 5 sqrt(6) ) ( "−" sqrt(12) )$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis −2 the 4th root of 18 close parenthesis open parenthesis − the 4th root of 9 close parenthesis")[$( −2 root(4, 18) ) ( "−" root(4, 9) )$]

ⓐ #math.equation(block: false, alt: "open parenthesis 3 the square root of 2 x cubed close parenthesis open parenthesis 7 the square root of 18 x squared close parenthesis")[$( 3 sqrt(2 x^(3)) ) ( 7 sqrt(18 x^(2)) )$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis −6 the cube root of 20 a squared close parenthesis open parenthesis −2 the cube root of 16 a cubed close parenthesis")[$( −6 root(3, 20 a^(2)) ) ( −2 root(3, 16 a^(3)) )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "126 x squared the square root of x")[$126 x^(2) sqrt(x)$] ⓑ #math.equation(block: false, alt: "48 a the cube root of 5 a squared")[$48 a root(3, 5 a^(2))$]
]

#strong[Use Polynomial Multiplication to Multiply Radical Expressions]

In the following exercises, multiply.

ⓐ #math.equation(block: false, alt: "the square root of 11 open parenthesis 8 plus 4 the square root of 11 close parenthesis")[$sqrt(11) ( 8 + 4 sqrt(11) )$] #linebreak() ⓑ #math.equation(block: false, alt: "the cube root of 3 open parenthesis the cube root of 9 plus the cube root of 18 close parenthesis")[$root(3, 3) ( root(3, 9) + root(3, 18) )$]

ⓐ #math.equation(block: false, alt: "open parenthesis 3 minus 2 the square root of 7 close parenthesis open parenthesis 5 minus 4 the square root of 7 close parenthesis")[$( 3 − 2 sqrt(7) ) ( 5 − 4 sqrt(7) )$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis the cube root of x minus 5 close parenthesis open parenthesis the cube root of x minus 3 close parenthesis")[$( root(3, x) − 5 ) ( root(3, x) − 3 )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "71 minus 22 the square root of 7")[$71 − 22 sqrt(7)$] #linebreak() ⓑ #math.equation(block: false, alt: "the cube root of x squared minus 8 the cube root of x plus 15")[$root(3, x^(2)) − 8 root(3, x) + 15$]
]

#math.equation(block: true, alt: "open parenthesis 2 the square root of 7 minus 5 the square root of 11 close parenthesis open parenthesis 4 the square root of 7 plus 9 the square root of 11 close parenthesis")[$( 2 sqrt(7) − 5 sqrt(11) ) ( 4 sqrt(7) + 9 sqrt(11) )$]

ⓐ #math.equation(block: false, alt: "open parenthesis 4 plus the square root of 11 close parenthesis squared")[$attach(( 4 + sqrt(11) ), t: 2)$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis 3 minus 2 the square root of 5 close parenthesis squared")[$attach(( 3 − 2 sqrt(5) ), t: 2)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "27 plus 8 the square root of 11")[$27 + 8 sqrt(11)$] ⓑ #math.equation(block: false, alt: "29 minus 12 the square root of 5")[$29 − 12 sqrt(5)$]
]

#math.equation(block: true, alt: "open parenthesis 7 plus the square root of 10 close parenthesis open parenthesis 7 minus the square root of 10 close parenthesis")[$( 7 + sqrt(10) ) ( 7 − sqrt(10) )$]

#math.equation(block: true, alt: "open parenthesis the cube root of 3 x plus 2 close parenthesis open parenthesis the cube root of 3 x minus 2 close parenthesis")[$( root(3, 3 x) + 2 ) ( root(3, 3 x) − 2 )$]

#solutionbox[
#math.equation(block: true, alt: "the cube root of 9 x squared minus 4")[$root(3, 9 x^(2)) − 4$]
]


==== Divide Radical Expressions

#strong[Divide Square Roots]

In the following exercises, simplify.

ⓐ #math.equation(block: false, alt: "the fraction the square root of 48 over the square root of 75")[$frac(sqrt(48), sqrt(75))$] #linebreak() ⓑ #math.equation(block: false, alt: "the fraction the cube root of 81 over the cube root of 24")[$frac(root(3, 81), root(3, 24))$]

ⓐ #math.equation(block: false, alt: "the fraction the square root of 320 m n to the power −5 over the square root of 45 m to the power −7 n cubed")[$frac(sqrt(320 m n^(−5)), sqrt(45 m^(−7) n^(3)))$] #linebreak() ⓑ #math.equation(block: false, alt: "the fraction the cube root of 16 x to the power 4 y to the power −2 over the cube root of −54 x to the power −2 y to the power 4")[$frac(root(3, 16 x^(4) y^(−2)), root(3, −54 x^(−2) y^(4)))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction 8 m to the power 4 over 3 n to the power 4")[$frac(8 m^(4), 3 n^(4))$] ⓑ #math.equation(block: false, alt: "minus the fraction 2 x squared over 3 y squared")[$− frac(2 x^(2), 3 y^(2))$]
]

#strong[Rationalize a One Term Denominator]

In the following exercises, rationalize the denominator.

ⓐ #math.equation(block: false, alt: "the fraction 8 over the square root of 3")[$frac(8, sqrt(3))$] ⓑ #math.equation(block: false, alt: "the square root of the fraction 7 over 40")[$sqrt(frac(7, 40))$] ⓒ #math.equation(block: false, alt: "the fraction 8 over the square root of 2 y")[$frac(8, sqrt(2 y))$]

ⓐ #math.equation(block: false, alt: "the fraction 1 over the cube root of 11")[$frac(1, root(3, 11))$] ⓑ #math.equation(block: false, alt: "the cube root of the fraction 7 over 54")[$root(3, frac(7, 54))$] ⓒ #math.equation(block: false, alt: "the fraction 3 over the cube root of 3 x squared")[$frac(3, root(3, 3 x^(2)))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction the cube root of 121 over 11")[$frac(root(3, 121), 11)$] ⓑ #math.equation(block: false, alt: "the fraction the cube root of 28 over 6")[$frac(root(3, 28), 6)$] ⓒ #math.equation(block: false, alt: "the fraction the cube root of 9 x over x")[$frac(root(3, 9 x), x)$]
]

ⓐ #math.equation(block: false, alt: "the fraction 1 over the 4th root of 4")[$frac(1, root(4, 4))$] ⓑ #math.equation(block: false, alt: "the 4th root of the fraction 9 over 32")[$root(4, frac(9, 32))$] ⓒ #math.equation(block: false, alt: "the fraction 6 over the 4th root of 9 x cubed")[$frac(6, root(4, 9 x^(3)))$]

#strong[Rationalize a Two Term Denominator]

In the following exercises, simplify.

#math.equation(block: true, alt: "the fraction 7 over 2 minus the square root of 6")[$frac(7, 2 − sqrt(6))$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 7 open parenthesis 2 plus the square root of 6 close parenthesis over 2")[$− frac(7 ( 2 + sqrt(6) ), 2)$]
]

#math.equation(block: true, alt: "the fraction the square root of 5 over the square root of n minus the square root of 7")[$frac(sqrt(5), sqrt(n) − sqrt(7))$]

#math.equation(block: true, alt: "the fraction the square root of x plus the square root of 8 over the square root of x minus the square root of 8")[$frac(sqrt(x) + sqrt(8), sqrt(x) − sqrt(8))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction open parenthesis the square root of x plus 2 the square root of 2 close parenthesis over x minus 8 squared")[$attach(frac(( sqrt(x) + 2 sqrt(2) ), x − 8), t: 2)$]
]


==== Solve Radical Equations

#strong[Solve Radical Equations]

In the following exercises, solve.

#math.equation(block: true, alt: "the square root of 4 x minus 3 equals 7")[$sqrt(4 x − 3) = 7$]

#math.equation(block: true, alt: "the square root of 5 x plus 1 equals −3")[$sqrt(5 x + 1) = −3$]

#solutionbox[
no solution
]

#math.equation(block: true, alt: "the cube root of 4 x minus 1 equals 3")[$root(3, 4 x − 1) = 3$]

#math.equation(block: true, alt: "the square root of u minus 3 plus 3 equals u")[$sqrt(u − 3) + 3 = u$]

#solutionbox[
#math.equation(block: true, alt: "u equals 3 , u equals 4")[$u = 3 , u = 4$]
]

#math.equation(block: true, alt: "the cube root of 4 x plus 5 minus 2 equals −5")[$root(3, 4 x + 5) − 2 = −5$]

#math.equation(block: true, alt: "open parenthesis 8 x plus 5 close parenthesis to the power the fraction 1 over 3 plus 2 equals −1")[$attach(( 8 x + 5 ), t: frac(1, 3)) + 2 = −1$]

#solutionbox[
#math.equation(block: true, alt: "x equals −4")[$x = −4$]
]

#math.equation(block: true, alt: "the square root of y plus 4 minus y plus 2 equals 0")[$sqrt(y + 4) − y + 2 = 0$]

#math.equation(block: true, alt: "2 the square root of 8 r plus 1 minus 8 equals 2")[$2 sqrt(8 r + 1) − 8 = 2$]

#solutionbox[
#math.equation(block: true, alt: "r equals 3")[$r = 3$]
]

#strong[Solve Radical Equations with Two Radicals]

In the following exercises, solve.

#math.equation(block: true, alt: "the square root of 10 plus 2 c equals the square root of 4 c plus 16")[$sqrt(10 + 2 c) = sqrt(4 c + 16)$]

#math.equation(block: true, alt: "the cube root of 2 x squared plus 9 x minus 18 equals the cube root of x squared plus 3 x minus 2")[$root(3, 2 x^(2) + 9 x − 18) = root(3, x^(2) + 3 x − 2)$]

#solutionbox[
#math.equation(block: true, alt: "x equals −8 , x equals 2")[$x = −8 , x = 2$]
]

#math.equation(block: true, alt: "the square root of r plus 6 equals the square root of r plus 8")[$sqrt(r) + 6 = sqrt(r + 8)$]

#math.equation(block: true, alt: "the square root of x plus 1 minus the square root of x minus 2 equals 1")[$sqrt(x + 1) − sqrt(x − 2) = 1$]

#solutionbox[
#math.equation(block: true, alt: "x equals 3")[$x = 3$]
]

#strong[Use Radicals in Applications]

In the following exercises, solve. Round approximations to one decimal place.

#strong[Landscaping] Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of 75 square feet. Use the formula #math.equation(block: false, alt: "s equals the square root of A")[$s = sqrt(A)$] to find the length of each side of his garden. Round your answer to the nearest tenth of a foot.

#strong[Accident investigation] An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 175 feet. Use the formula #math.equation(block: false, alt: "s equals the square root of 24 d")[$s = sqrt(24 d)$] to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.

#solutionbox[
#math.equation(block: false, alt: "64.8")[$64.8$] feet
]


==== Use Radicals in Functions

#strong[Evaluate a Radical Function]

In the following exercises, evaluate each function.

#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the square root of 6 x plus 1 ,")[$g ( x ) = sqrt(6 x + 1) ,$] find #linebreak() ⓐ #math.equation(block: false, alt: "g open parenthesis 4 close parenthesis")[$g ( 4 )$] #linebreak() ⓑ #math.equation(block: false, alt: "g open parenthesis 8 close parenthesis")[$g ( 8 )$]

#math.equation(block: false, alt: "G open parenthesis x close parenthesis equals the square root of 5 x minus 1 ,")[$G ( x ) = sqrt(5 x − 1) ,$] find #linebreak() ⓐ #math.equation(block: false, alt: "G open parenthesis 5 close parenthesis")[$G ( 5 )$] #linebreak() ⓑ #math.equation(block: false, alt: "G open parenthesis 2 close parenthesis")[$G ( 2 )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "G open parenthesis 5 close parenthesis equals 2 the square root of 6")[$G ( 5 ) = 2 sqrt(6)$] ⓑ #math.equation(block: false, alt: "G open parenthesis 2 close parenthesis equals 3")[$G ( 2 ) = 3$]
]

#math.equation(block: false, alt: "h open parenthesis x close parenthesis equals the cube root of x squared minus 4 ,")[$h ( x ) = root(3, x^(2) − 4) ,$] find #linebreak() ⓐ #math.equation(block: false, alt: "h open parenthesis −2 close parenthesis")[$h ( −2 )$] #linebreak() ⓑ #math.equation(block: false, alt: "h open parenthesis 6 close parenthesis")[$h ( 6 )$]

For the function #linebreak() #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the 4th root of 4 minus 4 x ,")[$g ( x ) = root(4, 4 − 4 x) ,$] find #linebreak() ⓐ #math.equation(block: false, alt: "g open parenthesis 1 close parenthesis")[$g ( 1 )$] #linebreak() ⓑ #math.equation(block: false, alt: "g open parenthesis −3 close parenthesis")[$g ( −3 )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "g open parenthesis 1 close parenthesis equals 0")[$g ( 1 ) = 0$] ⓑ #math.equation(block: false, alt: "g open parenthesis −3 close parenthesis equals 2")[$g ( −3 ) = 2$]
]

#strong[Find the Domain of a Radical Function]

In the following exercises, find the domain of the function and write the domain in interval notation.

#math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the square root of 2 minus 3 x")[$g ( x ) = sqrt(2 − 3 x)$]

#math.equation(block: true, alt: "F open parenthesis x close parenthesis equals the square root of the fraction x plus 3 over x minus 2")[$F ( x ) = sqrt(frac(x + 3, x − 2))$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 2 , ∞ close parenthesis")[$( 2 , ∞ )$]
]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the cube root of 4 x squared minus 16")[$f ( x ) = root(3, 4 x^(2) − 16)$]

#math.equation(block: true, alt: "F open parenthesis x close parenthesis equals the 4th root of 10 minus 7 x")[$F ( x ) = root(4, 10 − 7 x)$]

#solutionbox[
#math.equation(block: true, alt: "minus ∞ , the fraction 10 over 7")[$− ∞ , frac(10, 7)$]
]

#strong[Graph Radical Functions]

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

#math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the square root of x plus 4")[$g ( x ) = sqrt(x + 4)$]

#math.equation(block: true, alt: "g open parenthesis x close parenthesis equals 2 the square root of x")[$g ( x ) = 2 sqrt(x)$]

#solutionbox[
ⓐ domain: #math.equation(block: false, alt: "[ 0 , ∞ close parenthesis")[$[ 0 , ∞ )$] #linebreak() 
ⓑ  #linebreak() #figure(figph[The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 8. The function has a starting point at (0, 0) and goes through the points (1, 2) and (4, 4).], alt: "The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 8. The function has a starting point at (0, 0) and goes through the points (1, 2) and (4, 4).", caption: none)

 #linebreak() 
ⓒ range: #math.equation(block: false, alt: "[ 0 , ∞ close parenthesis")[$[ 0 , ∞ )$]
]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the cube root of x minus 1")[$f ( x ) = root(3, x − 1)$]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the cube root of x plus 3")[$f ( x ) = root(3, x) + 3$]

#solutionbox[
ⓐ domain: #math.equation(block: false, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$] #linebreak() 
ⓑ  #linebreak() #figure(figph[The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The function has a center point at (0, 3) and goes through the points (negative 1, 2) and (1, 4).], alt: "The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The function has a center point at (0, 3) and goes through the points (negative 1, 2) and (1, 4).", caption: none)

 #linebreak() 
ⓒ range: #math.equation(block: false, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$]
]


==== Use the Complex Number System

#strong[Evaluate the Square Root of a Negative Number]

In the following exercises, write each expression in terms of #emph[i] and simplify if possible.

ⓐ #math.equation(block: false, alt: "the square root of −100")[$sqrt(−100)$] #linebreak() ⓑ #math.equation(block: false, alt: "the square root of −13")[$sqrt(−13)$] #linebreak() ⓒ #math.equation(block: false, alt: "the square root of −45")[$sqrt(−45)$]

#strong[Add or Subtract Complex Numbers]

In the following exercises, add or subtract.

#math.equation(block: true, alt: "the square root of −50 plus the square root of −18")[$sqrt(−50) + sqrt(−18)$]

#solutionbox[
#math.equation(block: true, alt: "8 the square root of 2 i")[$8 sqrt(2) i$]
]

#math.equation(block: true, alt: "open parenthesis 8 minus i close parenthesis plus open parenthesis 6 plus 3 i close parenthesis")[$( 8 − i ) + ( 6 + 3 i )$]

#math.equation(block: true, alt: "open parenthesis 6 plus i close parenthesis minus open parenthesis −2 minus 4 i close parenthesis")[$( 6 + i ) − ( −2 − 4 i )$]

#solutionbox[
#math.equation(block: true, alt: "8 plus 5 i")[$8 + 5 i$]
]

#math.equation(block: true, alt: "open parenthesis −7 minus the square root of −50 close parenthesis minus open parenthesis −32 minus the square root of −18 close parenthesis")[$( −7 − sqrt(−50) ) − ( −32 − sqrt(−18) )$]

#strong[Multiply Complex Numbers]

In the following exercises, multiply.

#math.equation(block: true, alt: "open parenthesis −2 minus 5 i close parenthesis open parenthesis −4 plus 3 i close parenthesis")[$( −2 − 5 i ) ( −4 + 3 i )$]

#solutionbox[
#math.equation(block: true, alt: "23 plus 14 i")[$23 + 14 i$]
]

#math.equation(block: true, alt: "−6 i open parenthesis −3 minus 2 i close parenthesis")[$−6 i ( −3 − 2 i )$]

#math.equation(block: true, alt: "the square root of −4 times the square root of −16")[$sqrt(−4) · sqrt(−16)$]

#solutionbox[
#math.equation(block: true, alt: "−8")[$−8$]
]

#math.equation(block: true, alt: "open parenthesis 5 minus the square root of −12 close parenthesis open parenthesis −3 plus the square root of −75 close parenthesis")[$( 5 − sqrt(−12) ) ( −3 + sqrt(−75) )$]

In the following exercises, multiply using the Product of Binomial Squares Pattern.

#math.equation(block: true, alt: "open parenthesis −2 minus 3 i close parenthesis squared")[$attach(( −2 − 3 i ), t: 2)$]

#solutionbox[
#math.equation(block: true, alt: "−5 plus 12 i")[$−5 + 12 i$]
]

In the following exercises, multiply using the Product of Complex Conjugates Pattern.

#math.equation(block: true, alt: "open parenthesis 9 minus 2 i close parenthesis open parenthesis 9 plus 2 i close parenthesis")[$( 9 − 2 i ) ( 9 + 2 i )$]

#strong[Divide Complex Numbers]

In the following exercises, divide.

#math.equation(block: true, alt: "the fraction 2 plus i over 3 minus 4 i")[$frac(2 + i, 3 − 4 i)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 over 25 plus the fraction 11 over 25 i")[$frac(2, 25) + frac(11, 25) i$]
]

#math.equation(block: true, alt: "the fraction −4 over 3 minus 2 i")[$frac(−4, 3 − 2 i)$]

#strong[Simplify Powers of #emph[i]]

In the following exercises, simplify.

#math.equation(block: true, alt: "i to the power 48")[$i^(48)$]

#solutionbox[
1
]

#math.equation(block: true, alt: "i to the power 255")[$i^(255)$]


=== Practice Test

In the following exercises, simplify using absolute values as necessary.

#math.equation(block: true, alt: "the cube root of 125 x to the power 9")[$root(3, 125 x^(9))$]

#solutionbox[
#math.equation(block: true, alt: "5 x cubed")[$5 x^(3)$]
]

#math.equation(block: true, alt: "the square root of 169 x to the power 8 y to the power 6")[$sqrt(169 x^(8) y^(6))$]

#math.equation(block: true, alt: "the cube root of 72 x to the power 8 y to the power 4")[$root(3, 72 x^(8) y^(4))$]

#solutionbox[
#math.equation(block: true, alt: "2 x squared y to the power the cube root of 9 x squared y")[$2 x^(2) y^ root(3, 9 x^(2) y)$]
]

#math.equation(block: true, alt: "the square root of the fraction 45 x cubed y to the power 4 over 180 x to the power 5 y squared")[$sqrt(frac(45 x^(3) y^(4), 180 x^(5) y^(2)))$]

In the following exercises, simplify. Assume all variables are positive.

ⓐ #math.equation(block: false, alt: "256 to the power minus the fraction 1 over 4")[$256^(− frac(1, 4))$] ⓑ #math.equation(block: false, alt: "− 49 to the power the fraction 3 over 2")[$"−" 49^(frac(3, 2))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] ⓑ #math.equation(block: false, alt: "−343")[$−343$]
]

#math.equation(block: true, alt: "the square root of −45")[$sqrt(−45)$]

#math.equation(block: true, alt: "the fraction x to the power minus the fraction 1 over 4 times x to the power the fraction 5 over 4 over x to the power minus the fraction 3 over 4")[$frac(x^(− frac(1, 4)) · x^(frac(5, 4)), x^(− frac(3, 4)))$]

#solutionbox[
#math.equation(block: true, alt: "x to the power the fraction 7 over 4")[$x^(frac(7, 4))$]
]

#math.equation(block: true, alt: "open parenthesis the fraction 8 ​ x to the power the fraction 2 over 3 ​ y to the power minus the fraction 5 over 2 over x to the power minus the fraction 7 over 3 y to the power the fraction 1 over 2 close parenthesis to the power the fraction 1 over 3")[$attach(( frac(8 "​" x^(frac(2, 3)) "​" y^(− frac(5, 2)), x^(− frac(7, 3)) y^(frac(1, 2))) ), t: frac(1, 3))$]

#math.equation(block: true, alt: "the square root of 48 x to the power 5 minus the square root of 75 x to the power 5")[$sqrt(48 x^(5)) − sqrt(75 x^(5))$]

#solutionbox[
#math.equation(block: true, alt: "− x squared the square root of 3 x")[$"−" x^(2) sqrt(3 x)$]
]

#math.equation(block: true, alt: "the square root of 27 x squared minus 4 x the square root of 12 plus the square root of 108 x squared")[$sqrt(27 x^(2)) − 4 x sqrt(12) + sqrt(108 x^(2))$]

#math.equation(block: true, alt: "2 the square root of 12 x to the power 5 times 3 the square root of 6 x cubed")[$2 sqrt(12 x^(5)) · 3 sqrt(6 x^(3))$]

#solutionbox[
#math.equation(block: true, alt: "36 x to the power 4 the square root of 2")[$36 x^(4) sqrt(2)$]
]

#math.equation(block: true, alt: "the cube root of 4 open parenthesis the cube root of 16 minus the cube root of 6 close parenthesis")[$root(3, 4) ( root(3, 16) − root(3, 6) )$]

#math.equation(block: true, alt: "open parenthesis 4 minus 3 the square root of 3 close parenthesis open parenthesis 5 plus 2 the square root of 3 close parenthesis")[$( 4 − 3 sqrt(3) ) ( 5 + 2 sqrt(3) )$]

#solutionbox[
#math.equation(block: true, alt: "2 minus 7 the square root of 3")[$2 − 7 sqrt(3)$]
]

#math.equation(block: true, alt: "the fraction the cube root of 128 over the cube root of 54")[$frac(root(3, 128), root(3, 54))$]

#math.equation(block: true, alt: "the fraction the square root of 245 x y to the power −4 over the square root of 45 x to the power −4 y cubed")[$frac(sqrt(245 x y^(−4)), sqrt(45 x^(−4) y^(3)))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 7 x squared the square root of x over 3 | y cubed | the square root of y")[$frac(7 x^(2) sqrt(x), 3 | y^(3) | sqrt(y))$]
]

#math.equation(block: true, alt: "the fraction 1 over the cube root of 5")[$frac(1, root(3, 5))$]

#math.equation(block: true, alt: "the fraction 3 over 2 plus the square root of 3")[$frac(3, 2 + sqrt(3))$]

#solutionbox[
#math.equation(block: true, alt: "3 open parenthesis 2 minus the square root of 3 close parenthesis")[$3 ( 2 − sqrt(3) )$]
]

#math.equation(block: true, alt: "the square root of −4 times the square root of −9")[$sqrt(−4) · sqrt(−9)$]

#math.equation(block: true, alt: "−4 i open parenthesis −2 minus 3 i close parenthesis")[$−4 i ( −2 − 3 i )$]

#solutionbox[
#math.equation(block: true, alt: "−12 plus 8 i")[$−12 + 8 i$]
]

#math.equation(block: true, alt: "the fraction 4 plus i over 3 minus 2 i")[$frac(4 + i, 3 − 2 i)$]

#math.equation(block: true, alt: "i to the power 172")[$i^(172)$]

#solutionbox[
#math.equation(block: true, alt: "1")[$1$]
]

In the following exercises, solve.

#math.equation(block: true, alt: "the square root of 2 x plus 5 plus 8 equals 6")[$sqrt(2 x + 5) + 8 = 6$]

#math.equation(block: true, alt: "the square root of x plus 5 plus 1 equals x")[$sqrt(x + 5) + 1 = x$]

#solutionbox[
#math.equation(block: true, alt: "x equals 4")[$x = 4$]
]

#math.equation(block: true, alt: "the cube root of 2 x squared minus 6 x minus 23 equals the cube root of x squared minus 3 x plus 5")[$root(3, 2 x^(2) − 6 x − 23) = root(3, x^(2) − 3 x + 5)$]

In the following exercise, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

#math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the square root of x plus 2")[$g ( x ) = sqrt(x + 2)$]

#solutionbox[
ⓐ domain: #math.equation(block: false, alt: "[ −2 , ∞ close parenthesis")[$[ −2 , ∞ )$] #linebreak() 
ⓑ  #linebreak() #figure(figph[The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 6. The y-axis runs from 0 to 8. The function has a starting point at (negative 2, 0) and goes through the points (negative 1, 1) and (2, 2).], alt: "The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 6. The y-axis runs from 0 to 8. The function has a starting point at (negative 2, 0) and goes through the points (negative 1, 1) and (2, 2).", caption: none)

 #linebreak() 
ⓒ range: #math.equation(block: false, alt: "[ 0 , ∞ close parenthesis")[$[ 0 , ∞ )$]
]