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== 10.2#h(0.6em)Use Multiplication Properties of Exponents

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

Simplify: #math.equation(block: false, alt: "the fraction 3 over 4 times the fraction 3 over 4 .")[$frac(3, 4) · frac(3, 4) .$] #linebreak() 
If you missed the problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 9 over 16")[$frac(9, 16)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis −2 close parenthesis open parenthesis −2 close parenthesis open parenthesis −2 close parenthesis .")[$( −2 ) ( −2 ) ( −2 ) .$] #linebreak() 
If you missed the problem, review .

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


=== Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, #math.equation(block: false, alt: "2 to the power 4")[$2^(4)$] means to multiply four factors of #math.equation(block: false, alt: "2 ,")[$2 ,$] so #math.equation(block: false, alt: "2 to the power 4")[$2^(4)$] means #math.equation(block: false, alt: "2 times 2 times 2 times 2 .")[$2 · 2 · 2 · 2 .$] This format is known as #strong[exponential notation].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Exponential Notation]

#figure(figph[On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.], alt: "On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.", caption: none)

This is read #math.equation(block: false, alt: "a")[$a$] to the #math.equation(block: false, alt: "m to the power th")[$m^("th")$] power.
]

In the expression #math.equation(block: false, alt: "a to the power m ,")[$a^(m) ,$] the #strong[exponent] tells us how many times we use the #strong[base] #math.equation(block: false, alt: "a")[$a$] as a factor.

#figure(figph[On the left side, 7 to the 3rd power is shown. Below is 7 times 7 times 7, with 3 factors written below. On the right side, parentheses negative 8 to the 5th power is shown. Below is negative 8 times negative 8 times negative 8 times negative 8 times negative 8, with 5 factors written below.], alt: "On the left side, 7 to the 3rd power is shown. Below is 7 times 7 times 7, with 3 factors written below. On the right side, parentheses negative 8 to the 5th power is shown. Below is negative 8 times negative 8 times negative 8 times negative 8 times negative 8, with 5 factors written below.", caption: none)

Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.

#examplebox("Example 1")[][
Simplify:

+ ⓐ #math.equation(block: false, alt: "5 cubed")[$#h(0.2em) 5^(3)$]
+ ⓑ #math.equation(block: false, alt: "9 to the power 1")[$#h(0.2em) 9^(1)$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "5 cubed")[$5^(3)$]],
  [Multiply 3 factors of 5.], [#math.equation(block: false, alt: "5 times 5 times 5")[$5 · 5 · 5$]],
  [Simplify.], [#math.equation(block: false, alt: "125")[$125$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "9 to the power 1")[$9^(1)$]],
  [Multiply 1 factor of 9.], [#math.equation(block: false, alt: "9")[$9$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

+ ⓐ #math.equation(block: false, alt: "4 cubed")[$#h(0.2em) 4^(3)$]
+ ⓑ #math.equation(block: false, alt: "11 to the power 1")[$#h(0.2em) 11^(1)$]

#solutionbox[
+ ⓐ 64
+ ⓑ 11
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

+ ⓐ #math.equation(block: false, alt: "3 to the power 4")[$#h(0.2em) 3^(4)$]
+ ⓑ #math.equation(block: false, alt: "21 to the power 1")[$#h(0.2em) 21^(1)$]

#solutionbox[
+ ⓐ 81
+ ⓐ 21
]
]

#examplebox("Example 2")[][
Simplify:

+ ⓐ #math.equation(block: false, alt: "open parenthesis the fraction 7 over 8 close parenthesis squared")[$#h(0.2em) attach(( frac(7, 8) ), t: 2)$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 0.74 close parenthesis squared")[$#h(0.2em) attach(( 0.74 ), t: 2)$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "open parenthesis the fraction 7 over 8 close parenthesis squared")[$attach(( frac(7, 8) ), t: 2)$]],
  [Multiply two factors.], [#math.equation(block: false, alt: "open parenthesis the fraction 7 over 8 close parenthesis open parenthesis the fraction 7 over 8 close parenthesis")[$( frac(7, 8) ) ( frac(7, 8) )$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 49 over 64")[$frac(49, 64)$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "open parenthesis 0.74 close parenthesis squared")[$attach(( 0.74 ), t: 2)$]],
  [Multiply two factors.], [#math.equation(block: false, alt: "open parenthesis 0.74 close parenthesis open parenthesis 0.74 close parenthesis")[$( 0.74 ) ( 0.74 )$]],
  [Simplify.], [#math.equation(block: false, alt: "0.5476")[$0.5476$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

+ ⓐ #math.equation(block: false, alt: "open parenthesis the fraction 5 over 8 close parenthesis squared")[$#h(0.2em) attach(( frac(5, 8) ), t: 2)$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 0.67 close parenthesis squared")[$#h(0.2em) attach(( 0.67 ), t: 2)$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 25 over 64")[$#h(0.2em) frac(25, 64)$]
+ ⓑ #math.equation(block: false, alt: "0.4489")[$#h(0.2em) 0.4489$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

+ ⓐ #math.equation(block: false, alt: "open parenthesis the fraction 2 over 5 close parenthesis cubed")[$#h(0.2em) attach(( frac(2, 5) ), t: 3)$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 0.127 close parenthesis squared")[$#h(0.2em) attach(( 0.127 ), t: 2)$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 8 over 125")[$#h(0.2em) frac(8, 125)$]
+ ⓑ #math.equation(block: false, alt: "0.016129")[$#h(0.2em) 0.016129$]
]
]

#examplebox("Example 3")[][
Simplify:

+ ⓐ #math.equation(block: false, alt: "open parenthesis −3 close parenthesis to the power 4")[$#h(0.2em) attach(( −3 ), t: 4)$]
+ ⓑ #math.equation(block: false, alt: "−3 to the power 4")[$#h(0.2em) −3^(4)$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "open parenthesis −3 close parenthesis to the power 4")[$attach(( −3 ), t: 4)$]],
  [Multiply four factors of −3.], [#math.equation(block: false, alt: "open parenthesis −3 close parenthesis open parenthesis −3 close parenthesis open parenthesis −3 close parenthesis open parenthesis −3 close parenthesis")[$( −3 ) ( −3 ) ( −3 ) ( −3 )$]],
  [Simplify.], [#math.equation(block: false, alt: "81")[$81$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "−3 to the power 4")[$−3^(4)$]],
  [Multiply two factors.], [#math.equation(block: false, alt: "minus open parenthesis 3 times 3 times 3 times 3 close parenthesis")[$− ( 3 · 3 · 3 · 3 )$]],
  [Simplify.], [#math.equation(block: false, alt: "−81")[$−81$]],
))

Notice the similarities and differences in parts ⓐ and ⓑ. Why are the answers different? In part ⓐ the parentheses tell us to raise the (−3) to the 4#super[th] power. In part ⓑ we raise only the 3 to the 4#super[th] power and then find the opposite.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

+ ⓐ #math.equation(block: false, alt: "open parenthesis −2 close parenthesis to the power 4")[$#h(0.2em) attach(( −2 ), t: 4)$]
+ ⓑ #math.equation(block: false, alt: "−2 to the power 4")[$#h(0.2em) −2^(4)$]

#solutionbox[
+ ⓐ 16
+ ⓑ −16
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

+ ⓐ #math.equation(block: false, alt: "open parenthesis −8 close parenthesis squared")[$#h(0.2em) attach(( −8 ), t: 2)$]
+ ⓑ #math.equation(block: false, alt: "−8 squared")[$#h(0.2em) −8^(2)$]

#solutionbox[
+ ⓐ 64
+ ⓑ −64
]
]


=== Simplify Expressions Using the Product Property of Exponents

You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too. We’ll derive the properties of exponents by looking for patterns in several examples. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.

First, we will look at an example that leads to the Product Property.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression shows 'x' raised to the power of 2, followed by a multiplication dot, and then 'x' raised to the power of 3. This can be simplified to x to the power of 5.], alt: "A mathematical expression shows 'x' raised to the power of 2, followed by a multiplication dot, and then 'x' raised to the power of 3. This can be simplified to x to the power of 5.", caption: none)]),
  [What does this mean?
  #linebreak()  #linebreak()  How many factors altogether?], [#figure(figph[Multiplying terms with the same base: (x\*x) has 2 factors, and (x\*x\*x) has 3 factors. Their product (x\*x\*x\*x\*x) has a total of 5 factors, demonstrating the addition of exponents.], alt: "Multiplying terms with the same base: (x*x) has 2 factors, and (x*x*x) has 3 factors. Their product (x*x*x*x*x) has a total of 5 factors, demonstrating the addition of exponents.", caption: none)],
  [So, we have], [#figure(figph[The mathematical expression x raised to the power of 5 (x^5) is centered on a plain white background.], alt: "The mathematical expression x raised to the power of 5 (x^5) is centered on a plain white background.", caption: none)],
  [Notice that 5 is the sum of the exponents, 2 and 3.], [#figure(figph[The image shows the mathematical expression x^2 multiplied by x^3, demonstrating that it equals x^(2+3), or x^5, illustrating the rule of adding exponents when multiplying powers with the same base.], alt: "The image shows the mathematical expression x^2 multiplied by x^3, demonstrating that it equals x^(2+3), or x^5, illustrating the rule of adding exponents when multiplying powers with the same base.", caption: none)],
  [We write:], [#math.equation(block: false, alt: "x squared times x cubed")[$x^(2) ⋅ x^(3)$] #linebreak() #math.equation(block: false, alt: "x to the power 2 plus 3")[$x^(2 + 3)$] #linebreak() #math.equation(block: false, alt: "x to the power 5")[$x^(5)$]],
))

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

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

If #math.equation(block: false, alt: "a")[$a$] is a real number and #math.equation(block: false, alt: "m , n")[$m , n$] are counting numbers, then

#math.equation(block: true, alt: "a to the power m times a to the power n equals a to the power m plus n")[$a^(m) · a^(n) = a^(m + n)$]To multiply with like bases, add the exponents.
]

An example with numbers helps to verify this property.

#math.equation(block: true, alt: "2 squared times 2 cubed, equals ?, 2 to the power 2 plus 3; 4 times 8, equals ?, 2 to the power 5; 32, equals, 32 ✓")[$2^(2) · 2^(3) & limits(=)^(?) & 2^(2 + 3) \ 4 · 8 & limits(=)^(?) & 2^(5) \ 32 & = & 32 ✓$]#examplebox("Example 4")[][
Simplify: #math.equation(block: false, alt: "x to the power 5 times x to the power 7 .")[$x^(5) · x^(7) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "x to the power 5 times x to the power 7")[$x^(5) · x^(7)$]]),
  [Use the product property, #math.equation(block: false, alt: "a to the power m times a to the power n equals a to the power m plus n .")[$a^(m) · a^(n) = a^(m + n) .$]], [#figure(figph[A mathematical expression showing 'x' raised to the power of '5+7', with the exponent '5+7' in red.], alt: "A mathematical expression showing 'x' raised to the power of '5+7', with the exponent '5+7' in red.", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "x to the power 12")[$x^(12)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "x to the power 7 times x to the power 8 .")[$x^(7) · x^(8) .$]

#solutionbox[
#emph[x]#super[15]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "x to the power 5 times x to the power 11 .")[$x^(5) · x^(11) .$]

#solutionbox[
#emph[x]#super[16]
]
]

#examplebox("Example 5")[][
Simplify: #math.equation(block: false, alt: "b to the power 4 times b .")[$b^(4) · b .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "b to the power 4 times b")[$b^(4) · b$]]),
  [Rewrite, #math.equation(block: false, alt: "b equals b to the power 1 .")[$b = b^(1) .$]], [#math.equation(block: false, alt: "b to the power 4 times b to the power 1")[$b^(4) · b^(1)$]],
  [Use the product property, #math.equation(block: false, alt: "a to the power m times a to the power n equals a to the power m plus n .")[$a^(m) · a^(n) = a^(m + n) .$]], [#figure(figph[A black lowercase letter 'b' with a red superscript '4+1', representing the mathematical expression b^(4+1).], alt: "A black lowercase letter 'b' with a red superscript '4+1', representing the mathematical expression b^(4+1).", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "b to the power 5")[$b^(5)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "p to the power 9 times p .")[$p^(9) · p .$]

#solutionbox[
#emph[p]#super[10]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "m times m to the power 7 .")[$m · m^(7) .$]

#solutionbox[
#emph[m]#super[8]
]
]

#examplebox("Example 6")[][
Simplify: #math.equation(block: false, alt: "2 to the power 7 times 2 to the power 9 .")[$2^(7) · 2^(9) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "2 to the power 7 times 2 to the power 9")[$2^(7) · 2^(9)$]]),
  [Use the product property, #math.equation(block: false, alt: "a to the power m times a to the power n equals a to the power m plus n .")[$a^(m) · a^(n) = a^(m + n) .$]], [#figure(figph[The mathematical expression 2 with an exponent of 7 plus 9, where the exponent 7+9 is rendered in red, indicating a potential highlight or distinction from the base number 2 which is in black.], alt: "The mathematical expression 2 with an exponent of 7 plus 9, where the exponent 7+9 is rendered in red, indicating a potential highlight or distinction from the base number 2 which is in black.", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "2 to the power 16")[$2^(16)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "6 times 6 to the power 9 .")[$6 · 6^(9) .$]

#solutionbox[
6#super[10]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "9 to the power 6 times 9 to the power 9 .")[$9^(6) · 9^(9) .$]

#solutionbox[
9#super[15]
]
]

#examplebox("Example 7")[][
Simplify: #math.equation(block: false, alt: "y to the power 17 times y to the power 23 .")[$y^(17) · y^(23) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "y to the power 17 times y to the power 23")[$y^(17) · y^(23)$]]),
  [Notice, the bases are the same, so add the exponents.], [#figure(figph[The mathematical expression y^(17+23) is displayed on a white background, featuring a black 'y' with a red exponent that includes the numbers 17 and 23 separated by a plus sign.], alt: "The mathematical expression y^(17+23) is displayed on a white background, featuring a black 'y' with a red exponent that includes the numbers 17 and 23 separated by a plus sign.", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "y to the power 40")[$y^(40)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "y to the power 24 times y to the power 19 .")[$y^(24) · y^(19) .$]

#solutionbox[
#emph[y]#super[43]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "z to the power 15 times z to the power 24 .")[$z^(15) · z^(24) .$]

#solutionbox[
#emph[z]#super[39]
]
]

We can extend the #strong[Product Property of Exponents] to more than two factors.

#examplebox("Example 8")[][
Simplify: #math.equation(block: false, alt: "x cubed times x to the power 4 times x squared .")[$x^(3) · x^(4) · x^(2) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "x cubed times x to the power 4 times x squared")[$x^(3) · x^(4) · x^(2)$]]),
  [Add the exponents, since the bases are the same.], [#figure(figph[A mathematical expression shows 'x' raised to the power of '3+4+2', with the exponent partially in red.], alt: "A mathematical expression shows 'x' raised to the power of '3+4+2', with the exponent partially in red.", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "x to the power 9")[$x^(9)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "x to the power 7 times x to the power 5 times x to the power 9 .")[$x^(7) · x^(5) · x^(9) .$]

#solutionbox[
#emph[x]#super[21]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "y cubed times y to the power 8 times y to the power 4 .")[$y^(3) · y^(8) · y^(4) .$]

#solutionbox[
#emph[y]#super[15]
]
]


=== Simplify Expressions Using the Power Property of Exponents

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing x squared raised to the power of 3, written as (x^2)^3.], alt: "A mathematical expression showing x squared raised to the power of 3, written as (x^2)^3.", caption: none)]),
  [], [#figure(figph[The mathematical expression 'x^2 . x^2 . x^2' is displayed, illustrating the multiplication of x squared by itself three times.], alt: "The mathematical expression 'x^2 . x^2 . x^2' is displayed, illustrating the multiplication of x squared by itself three times.", caption: none)],
  [What does this mean?
  #linebreak()  #linebreak()  How many factors altogether?], [#figure(figph[Illustration of factors: Three sets of 'x \* x', each labeled '2 factors', combine to show a total of '6 factors' in an algebraic expression.], alt: "Illustration of factors: Three sets of 'x * x', each labeled '2 factors', combine to show a total of '6 factors' in an algebraic expression.", caption: none)],
  [So, we have], [#figure(figph[The mathematical expression x^6 is displayed in black text on a plain white background, centered within the frame.], alt: "The mathematical expression x^6 is displayed in black text on a plain white background, centered within the frame.", caption: none)],
  [Notice that 6 is the product of the exponents, 2 and 3.], [#figure(figph[A mathematical expression states that (x^2)^3 is equal to x^(2\*3) or x^6, demonstrating the power of a power rule in exponents.], alt: "A mathematical expression states that (x^2)^3 is equal to x^(2*3) or x^6, demonstrating the power of a power rule in exponents.", caption: none)],
  [We write:], [#math.equation(block: false, alt: "open parenthesis x squared close parenthesis cubed")[$attach(( x^(2) ), t: 3)$] #linebreak() #math.equation(block: false, alt: "x to the power 2 times 3")[$x^(2 ⋅ 3)$] #linebreak() #math.equation(block: false, alt: "x to the power 6")[$x^(6)$]],
))

We multiplied the exponents. This leads to the Power Property for Exponents.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Power Property of Exponents]

If #math.equation(block: false, alt: "a")[$a$] is a real number and #math.equation(block: false, alt: "m , n")[$m , n$] are whole numbers, then

#math.equation(block: true, alt: "open parenthesis a to the power m close parenthesis to the power n equals a to the power m times n")[$attach(( a^(m) ), t: n) = a^(m · n)$]To raise a power to a power, multiply the exponents.
]

An example with numbers helps to verify this property.

#math.equation(block: true, alt: "open parenthesis 5 squared close parenthesis cubed, equals ?, 5 to the power 2 times 3; open parenthesis 25 close parenthesis cubed, equals ?, 5 to the power 6; 15,625, equals, 15,625 ✓")[$attach(( 5^(2) ), t: 3) & limits(=)^(?) & 5^(2 · 3) \ attach(( 25 ), t: 3) & limits(=)^(?) & 5^(6) \ 15,625 & = & 15,625 ✓$]#examplebox("Example 9")[][
Simplify:

+ ⓐ #math.equation(block: false, alt: "open parenthesis x to the power 5 close parenthesis to the power 7")[$#h(0.2em) attach(( x^(5) ), t: 7)$]
+ ⓐ #math.equation(block: false, alt: "open parenthesis 3 to the power 6 close parenthesis to the power 8")[$#h(0.2em) attach(( 3^(6) ), t: 8)$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "open parenthesis x to the power 5 close parenthesis to the power 7")[$attach(( x^(5) ), t: 7)$]],
  [Use the Power Property, #math.equation(block: false, alt: "open parenthesis a to the power m close parenthesis to the power n equals a to the power m times n .")[$attach(( a^(m) ), t: n) = a^(m · n) .$]], [#figure(figph[A mathematical expression showing the variable 'x' raised to the power of '5.7', where the exponent '5.7' is distinctively colored in red.], alt: "A mathematical expression showing the variable 'x' raised to the power of '5.7', where the exponent '5.7' is distinctively colored in red.", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "x to the power 35")[$x^(35)$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "open parenthesis 3 to the power 6 close parenthesis to the power 8")[$attach(( 3^(6) ), t: 8)$]],
  [Use the Power Property, #math.equation(block: false, alt: "open parenthesis a to the power m close parenthesis to the power n equals a to the power m times n .")[$attach(( a^(m) ), t: n) = a^(m · n) .$]], [#figure(figph[The mathematical expression 3 to the power of 6.8.], alt: "The mathematical expression 3 to the power of 6.8.", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "3 to the power 48")[$3^(48)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

+ ⓐ #math.equation(block: false, alt: "open parenthesis x to the power 7 close parenthesis to the power 4")[$#h(0.2em) attach(( x^(7) ), t: 4)$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 7 to the power 4 close parenthesis to the power 8")[$#h(0.2em) attach(( 7^(4) ), t: 8)$]

#solutionbox[
+ ⓐ #emph[x]#super[28]
+ ⓑ 7#super[32]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

+ ⓐ #math.equation(block: false, alt: "open parenthesis x to the power 6 close parenthesis to the power 9")[$#h(0.2em) attach(( x^(6) ), t: 9)$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 8 to the power 6 close parenthesis to the power 7")[$#h(0.2em) attach(( 8^(6) ), t: 7)$]

#solutionbox[
+ ⓐ #emph[x]#super[54]
+ ⓑ 8#super[42]
]
]


=== Simplify Expressions Using the Product to a Power Property

We will now look at an expression containing a product that is raised to a power. Look for a pattern.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 2 x close parenthesis cubed")[$attach(( 2 x ), t: 3)$]]),
  [What does this mean?], [#math.equation(block: false, alt: "2 x times 2 x times 2 x")[$2 x · 2 x · 2 x$]],
  [We group the like factors together.], [#math.equation(block: false, alt: "2 times 2 times 2 times x times x times x")[$2 · 2 · 2 · x · x · x$]],
  [How many factors of 2 and of #math.equation(block: false, alt: "x ?")[$x ?$]], [#math.equation(block: false, alt: "2 cubed times x cubed")[$2^(3) · x^(3)$]],
  [Notice that each factor was raised to the power.], [#math.equation(block: false, alt: "open parenthesis 2 x close parenthesis cubed is 2 cubed times x cubed")[$attach(( 2 x ), t: 3) #h(0.2em) "is" #h(0.2em) 2^(3) · x^(3)$]],
  [We write:], [#math.equation(block: false, alt: "open parenthesis 2 x close parenthesis cubed")[$attach(( 2 x ), t: 3)$]
    #linebreak() #math.equation(block: false, alt: "2 cubed times x cubed")[$2^(3) · x^(3)$]],
))

The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Product to a Power Property of Exponents]

If #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] are real numbers and #math.equation(block: false, alt: "m")[$m$] is a whole number, then

#math.equation(block: true, alt: "open parenthesis a b close parenthesis to the power m equals a to the power m b to the power m")[$attach(( a b ), t: m) = a^(m) b^(m)$]To raise a product to a power, raise each factor to that power.
]

An example with numbers helps to verify this property:

#math.equation(block: true, alt: "open parenthesis 2 times 3 close parenthesis squared, equals ?, 2 squared times 3 squared; 6 squared, equals ?, 4 times 9; 36, equals, 36 ✓")[$attach(( 2 · 3 ), t: 2) & limits(=)^(?) & 2^(2) · 3^(2) \ 6^(2) & limits(=)^(?) & 4 · 9 \ 36 & = & 36 ✓$]#examplebox("Example 10")[][
Simplify: #math.equation(block: false, alt: "open parenthesis −11 x close parenthesis squared .")[$attach(( −11 x ), t: 2) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis −11 x close parenthesis squared")[$attach(( −11 x ), t: 2)$]]),
  [Use the Power of a Product Property, #math.equation(block: false, alt: "open parenthesis a b close parenthesis to the power m equals a to the power m b to the power m .")[$attach(( a b ), t: m) = a^(m) b^(m) .$]], [#figure(figph[A mathematical expression is shown, consisting of an opening parenthesis, a minus sign, the number 11, a closing parenthesis, a superscript 2, the letter x, and another superscript 2. The superscripts are in red.], alt: "A mathematical expression is shown, consisting of an opening parenthesis, a minus sign, the number 11, a closing parenthesis, a superscript 2, the letter x, and another superscript 2. The superscripts are in red.", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "121 x squared")[$121 x^(2)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis −14 x close parenthesis squared .")[$attach(( −14 x ), t: 2) .$]

#solutionbox[
196#emph[x]#super[2]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis −12 a close parenthesis squared .")[$attach(( −12 a ), t: 2) .$]

#solutionbox[
144#emph[a]#super[2]
]
]

#examplebox("Example 11")[][
Simplify: #math.equation(block: false, alt: "open parenthesis 3 x y close parenthesis cubed .")[$attach(( 3 x y ), t: 3) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 3 x y close parenthesis cubed")[$attach(( 3 x y ), t: 3)$]]),
  [Raise each factor to the third power.], [#figure(figph[A mathematical expression showing '3x^3y^3', where the exponents for 'x' and 'y' are red threes.], alt: "A mathematical expression showing '3x^3y^3', where the exponents for 'x' and 'y' are red threes.", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "27 x cubed y cubed")[$27 x^(3) y^(3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis −4 x y close parenthesis to the power 4 .")[$attach(( −4 x y ), t: 4) .$]

#solutionbox[
256#emph[x]#super[4]#emph[y]#super[4]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 6 x y close parenthesis cubed .")[$attach(( 6 x y ), t: 3) .$]

#solutionbox[
216#emph[x]#super[3]#emph[y]#super[3]
]
]


=== Simplify Expressions by Applying Several Properties

We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Properties of Exponents]

If #math.equation(block: false, alt: "a , b")[$a , b$] are real numbers and #math.equation(block: false, alt: "m , n")[$m , n$] are whole numbers, then

#math.equation(block: true, alt: "Product Property, a to the power m times a to the power n equals a to the power m plus n; Power Property, open parenthesis a to the power m close parenthesis to the power n equals a to the power m times n ; Product to a Power Property, open parenthesis a b close parenthesis to the power m equals a to the power m b to the power m")[$"Product Property" &  &  & #h(0.4em) a^(m) · a^(n) = a^(m + n) \ "Power Property" &  &  & #h(0.8em) attach(( a^(m) ), t: n) = a^(m · n) #h(0.3em) \ "Product to a Power Property" &  &  & #h(0.8em) attach(( a b ), t: m) = a^(m) b^(m)$]
]

#examplebox("Example 12")[][
Simplify: #math.equation(block: false, alt: "open parenthesis x squared close parenthesis to the power 6 open parenthesis x to the power 5 close parenthesis to the power 4 .")[$attach(( x^(2) ), t: 6) attach(( x^(5) ), t: 4) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis x squared close parenthesis to the power 6 open parenthesis x to the power 5 close parenthesis to the power 4")[$attach(( x^(2) ), t: 6) attach(( x^(5) ), t: 4)$]]),
  [Use the Power Property.], [#math.equation(block: false, alt: "x to the power 12 times x to the power 20")[$x^(12) · x^(20)$]],
  [Add the exponents.], [#math.equation(block: false, alt: "x to the power 32")[$x^(32)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis x to the power 4 close parenthesis cubed open parenthesis x to the power 7 close parenthesis to the power 4 .")[$attach(( x^(4) ), t: 3) attach(( x^(7) ), t: 4) .$]

#solutionbox[
#emph[x]#super[40]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis y to the power 9 close parenthesis squared open parenthesis y to the power 8 close parenthesis cubed .")[$attach(( y^(9) ), t: 2) attach(( y^(8) ), t: 3) .$]

#solutionbox[
#emph[y]#super[42]
]
]

#examplebox("Example 13")[][
Simplify: #math.equation(block: false, alt: "open parenthesis minus 7 x cubed y to the power 4 close parenthesis squared .")[$attach(( − 7 x^(3) y^(4) ), t: 2) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis minus 7 x cubed y to the power 4 close parenthesis squared")[$attach(( − 7 x^(3) y^(4) ), t: 2)$]]),
  [Take each factor to the second power.], [#math.equation(block: false, alt: "open parenthesis −7 close parenthesis squared open parenthesis x cubed close parenthesis squared open parenthesis y to the power 4 close parenthesis squared")[$attach(( −7 ), t: 2) attach(( x^(3) ), t: 2) attach(( y^(4) ), t: 2)$]],
  [Use the Power Property.], [#math.equation(block: false, alt: "49 x to the power 6 y to the power 8")[$49 x^(6) y^(8)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis minus 8 x to the power 4 y to the power 7 close parenthesis cubed .")[$attach(( − 8 x^(4) y^(7) ), t: 3) .$]

#solutionbox[
−512#emph[x]#super[12]#emph[y]#super[21]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis minus 3 a to the power 5 b to the power 6 close parenthesis to the power 4 .")[$attach(( − 3 a^(5) b^(6) ), t: 4) .$]

#solutionbox[
81#emph[a]#super[20]#emph[b]#super[24]
]
]

#examplebox("Example 14")[][
Simplify: #math.equation(block: false, alt: "open parenthesis 6 n close parenthesis squared open parenthesis 4 n cubed close parenthesis .")[$attach(( 6 n ), t: 2) ( 4 n^(3) ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 6 n close parenthesis squared open parenthesis 4 n cubed close parenthesis")[$attach(( 6 n ), t: 2) ( 4 n^(3) )$]]),
  [Raise #math.equation(block: false, alt: "6 n")[$6 n$] to the second power.], [#math.equation(block: false, alt: "6 squared n squared times 4 n cubed")[$6^(2) n^(2) · 4 n^(3)$]],
  [Simplify.], [#math.equation(block: false, alt: "36 n squared times 4 n cubed")[$36 n^(2) · 4 n^(3)$]],
  [Use the Commutative Property.], [#math.equation(block: false, alt: "36 times 4 times n squared times n cubed")[$36 · 4 · n^(2) · n^(3)$]],
  [Multiply the constants and add the exponents.], [#math.equation(block: false, alt: "144 n to the power 5")[$144 n^(5)$]],
))

Notice that in the first monomial, the exponent was outside the parentheses and it applied to both factors inside. In the second monomial, the exponent was inside the parentheses and so it only applied to the #emph[n].
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 7 n close parenthesis squared open parenthesis 2 n to the power 12 close parenthesis .")[$attach(( 7 n ), t: 2) ( 2 n^(12) ) .$]

#solutionbox[
98#emph[n]#super[14]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 4 m close parenthesis squared open parenthesis 3 m cubed close parenthesis .")[$attach(( 4 m ), t: 2) ( 3 m^(3) ) .$]

#solutionbox[
48#emph[m]#super[5]
]
]

#examplebox("Example 15")[][
Simplify: #math.equation(block: false, alt: "open parenthesis 3 p squared q close parenthesis to the power 4 open parenthesis 2 p q squared close parenthesis cubed .")[$attach(( 3 p^(2) q ), t: 4) attach(( 2 p q^(2) ), t: 3) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 3 p squared q close parenthesis to the power 4 open parenthesis 2 p q squared close parenthesis cubed")[$attach(( 3 p^(2) q ), t: 4) attach(( 2 p q^(2) ), t: 3)$]]),
  [Use the Power of a Product Property.], [#math.equation(block: false, alt: "3 to the power 4 open parenthesis p squared close parenthesis to the power 4 q to the power 4 times 2 cubed p cubed open parenthesis q squared close parenthesis cubed")[$3^(4) attach(( p^(2) ), t: 4) q^(4) · 2^(3) p^(3) attach(( q^(2) ), t: 3)$]],
  [Use the Power Property.], [#math.equation(block: false, alt: "81 p to the power 8 q to the power 4 times 8 p cubed q to the power 6")[$81 p^(8) q^(4) · 8 p^(3) q^(6)$]],
  [Use the Commutative Property.], [#math.equation(block: false, alt: "81 times 8 times p to the power 8 times p cubed times q to the power 4 times q to the power 6")[$81 · 8 · p^(8) · p^(3) · q^(4) · q^(6)$]],
  [Multiply the constants and add the exponents for  #linebreak() each variable.], [#math.equation(block: false, alt: "648 p to the power 11 q to the power 10")[$648 p^(11) q^(10)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis u cubed v squared close parenthesis to the power 5 open parenthesis 4 u v to the power 4 close parenthesis cubed .")[$attach(( u^(3) v^(2) ), t: 5) attach(( 4 u v^(4) ), t: 3) .$]

#solutionbox[
64#emph[u]#super[18]#emph[v]#super[22]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 5 x squared y cubed close parenthesis squared open parenthesis 3 x y to the power 4 close parenthesis cubed .")[$attach(( 5 x^(2) y^(3) ), t: 2) attach(( 3 x y^(4) ), t: 3) .$]

#solutionbox[
675#emph[x]#super[7]#emph[y]#super[18]
]
]


=== Multiply Monomials

Since a #strong[monomial] is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.

#examplebox("Example 16")[][
Multiply: #math.equation(block: false, alt: "open parenthesis 4 x squared close parenthesis open parenthesis minus 5 x cubed close parenthesis .")[$( 4 x^(2) ) ( − 5 x^(3) ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 4 x squared close parenthesis open parenthesis minus 5 x cubed close parenthesis")[$( 4 x^(2) ) ( − 5 x^(3) )$]]),
  [Use the Commutative Property to rearrange the factors.], [#math.equation(block: false, alt: "4 times open parenthesis −5 close parenthesis times x squared times x cubed")[$4 · ( −5 ) · x^(2) · x^(3)$]],
  [Multiply.], [#math.equation(block: false, alt: "minus 20 x to the power 5")[$− 20 x^(5)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 7 x to the power 7 close parenthesis open parenthesis minus 8 x to the power 4 close parenthesis .")[$( 7 x^(7) ) ( − 8 x^(4) ) .$]

#solutionbox[
−56#emph[x]#super[11]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis minus 9 y to the power 4 close parenthesis open parenthesis minus 6 y to the power 5 close parenthesis .")[$( − 9 y^(4) ) ( − 6 y^(5) ) .$]

#solutionbox[
54#emph[y]#super[9]
]
]

#examplebox("Example 17")[][
Multiply: #math.equation(block: false, alt: "open parenthesis the fraction 3 over 4 c cubed d close parenthesis open parenthesis 12 c d squared close parenthesis .")[$( frac(3, 4) #h(0.1em) c^(3) d ) ( 12 c d^(2) ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis the fraction 3 over 4 c cubed d close parenthesis open parenthesis 12 c d squared close parenthesis")[$( frac(3, 4) #h(0.1em) c^(3) d ) ( 12 c d^(2) )$]]),
  [Use the Commutative Property to rearrange  #linebreak() the factors.], [#math.equation(block: false, alt: "the fraction 3 over 4 times 12 times c cubed times c times d times d squared")[$frac(3, 4) · 12 · c^(3) · c · d · d^(2)$]],
  [Multiply.], [#math.equation(block: false, alt: "9 c to the power 4 d cubed")[$9 c^(4) d^(3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis the fraction 4 over 5 m to the power 4 n cubed close parenthesis open parenthesis 15 m n cubed close parenthesis .")[$( frac(4, 5) #h(0.1em) m^(4) n^(3) ) ( 15 m n^(3) ) .$]

#solutionbox[
12#emph[m]#super[5]#emph[n]#super[6]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis the fraction 2 over 3 p to the power 5 q close parenthesis open parenthesis 18 p to the power 6 q to the power 7 close parenthesis .")[$( frac(2, 3) #h(0.1em) p^(5) q ) ( 18 p^(6) q^(7) ) .$]

#solutionbox[
12#emph[p]#super[11]#emph[q]#super[8]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[ACCESS ADDITIONAL ONLINE RESOURCES]

- #link("https://www.openstax.org/l/24expproperties")[Exponent Properties]
- #link("https://www.openstax.org/l/expproperties2")[Exponent Properties 2]
]


=== Key Concepts

- #emph[Exponential Notation]
#figure(figph[On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.], alt: "On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.", caption: none)


This is read #math.equation(block: false, alt: "a")[$a$] to the #math.equation(block: false, alt: "m to the power th")[$m^("th")$] power.
- #emph[Product Property of Exponents]

- If #math.equation(block: false, alt: "a")[$a$] is a real number and #math.equation(block: false, alt: "m , n")[$m , n$] are counting numbers, then

#math.equation(block: true, alt: "a to the power m times a to the power n equals a to the power m plus n")[$a^(m) · a^(n) = a^(m + n)$]
- To multiply with like bases, add the exponents.
- #emph[Power Property for Exponents]
- If #math.equation(block: false, alt: "a")[$a$] is a real number and #math.equation(block: false, alt: "m , n")[$m , n$] are counting numbers, then #math.equation(block: true, alt: "open parenthesis a to the power m close parenthesis to the power n equals a to the power m times n")[$attach(( a^(m) ), t: n) = a^(m ⋅ n)$]
- #emph[Product to a Power Property for Exponents]
- If #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] are real numbers and #math.equation(block: false, alt: "m")[$m$] is a whole number, then#math.equation(block: true, alt: "open parenthesis a b close parenthesis to the power m equals a to the power m b to the power m")[$attach(( a b ), t: m) = a^(m) b^(m)$]


==== Practice Makes Perfect

#strong[Simplify Expressions with Exponents]

In the following exercises, simplify each expression with exponents.

#math.equation(block: true, alt: "4 to the power 5")[$4^(5)$]

#solutionbox[
1,024
]

#math.equation(block: true, alt: "10 cubed")[$10^(3)$]

#math.equation(block: true, alt: "open parenthesis the fraction 1 over 2 close parenthesis squared")[$attach(( frac(1, 2) ), t: 2)$]

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

#math.equation(block: true, alt: "open parenthesis the fraction 3 over 5 close parenthesis squared")[$attach(( frac(3, 5) ), t: 2)$]

#math.equation(block: true, alt: "open parenthesis 0.2 close parenthesis cubed")[$attach(( 0.2 ), t: 3)$]

#solutionbox[
0.008
]

#math.equation(block: true, alt: "open parenthesis 0.4 close parenthesis cubed")[$attach(( 0.4 ), t: 3)$]

#math.equation(block: true, alt: "open parenthesis −5 close parenthesis to the power 4")[$attach(( −5 ), t: 4)$]

#solutionbox[
625
]

#math.equation(block: true, alt: "open parenthesis −3 close parenthesis to the power 5")[$attach(( −3 ), t: 5)$]

#math.equation(block: true, alt: "−5 to the power 4")[$−5^(4)$]

#solutionbox[
−625
]

#math.equation(block: true, alt: "−3 to the power 5")[$−3^(5)$]

#math.equation(block: true, alt: "−10 to the power 4")[$−10^(4)$]

#solutionbox[
−10,000
]

#math.equation(block: true, alt: "−2 to the power 6")[$−2^(6)$]

#math.equation(block: true, alt: "open parenthesis minus the fraction 2 over 3 close parenthesis cubed")[$attach(( − frac(2, 3) ), t: 3)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 8 over 27")[$− frac(8, 27)$]
]

#math.equation(block: true, alt: "open parenthesis minus the fraction 1 over 4 close parenthesis to the power 4")[$attach(( − frac(1, 4) ), t: 4)$]

#math.equation(block: true, alt: "minus 0.5 squared")[$− 0.5^(2)$]

#solutionbox[
−0.25
]

#math.equation(block: true, alt: "minus 0.1 to the power 4")[$− 0.1^(4)$]

#strong[Simplify Expressions Using the Product Property of Exponents]

In the following exercises, simplify each expression using the Product Property of Exponents.

#math.equation(block: true, alt: "x cubed times x to the power 6")[$x^(3) · x^(6)$]

#solutionbox[
#emph[x]#super[9]
]

#math.equation(block: true, alt: "m to the power 4 times m squared")[$m^(4) · m^(2)$]

#math.equation(block: true, alt: "a times a to the power 4")[$a · a^(4)$]

#solutionbox[
#emph[a]#super[5]
]

#math.equation(block: true, alt: "y to the power 12 times y")[$y^(12) · y$]

#math.equation(block: true, alt: "3 to the power 5 times 3 to the power 9")[$3^(5) · 3^(9)$]

#solutionbox[
3#super[14]
]

#math.equation(block: true, alt: "5 to the power 10 times 5 to the power 6")[$5^(10) · 5^(6)$]

#math.equation(block: true, alt: "z times z squared times z cubed")[$z · z^(2) · z^(3)$]

#solutionbox[
#emph[z]#super[6]
]

#math.equation(block: true, alt: "a times a cubed times a to the power 5")[$a · a^(3) · a^(5)$]

#math.equation(block: true, alt: "x to the power a times x squared")[$x^(a) · x^(2)$]

#solutionbox[
#emph[x]#super[#emph[a]+2]
]

#math.equation(block: true, alt: "y to the power p times y cubed")[$y^(p) · y^(3)$]

#math.equation(block: true, alt: "y to the power a times y to the power b")[$y^(a) · y^(b)$]

#solutionbox[
#emph[y]#super[#emph[a]+#emph[b]]
]

#math.equation(block: true, alt: "x to the power p times x to the power q")[$x^(p) · x^(q)$]

#strong[Simplify Expressions Using the Power Property of Exponents]

In the following exercises, simplify each expression using the Power Property of Exponents#emph[.]

#math.equation(block: true, alt: "open parenthesis u to the power 4 close parenthesis squared")[$attach(( u^(4) ), t: 2)$]

#solutionbox[
#emph[u]#super[8]
]

#math.equation(block: true, alt: "open parenthesis x squared close parenthesis to the power 7")[$attach(( x^(2) ), t: 7)$]

#math.equation(block: true, alt: "open parenthesis y to the power 5 close parenthesis to the power 4")[$attach(( y^(5) ), t: 4)$]

#solutionbox[
#emph[y]#super[20]
]

#math.equation(block: true, alt: "open parenthesis a cubed close parenthesis squared")[$attach(( a^(3) ), t: 2)$]

#math.equation(block: true, alt: "open parenthesis 10 squared close parenthesis to the power 6")[$attach(( 10^(2) ), t: 6)$]

#solutionbox[
10#super[12]
]

#math.equation(block: true, alt: "open parenthesis 2 to the power 8 close parenthesis cubed")[$attach(( 2^(8) ), t: 3)$]

#math.equation(block: true, alt: "open parenthesis x to the power 15 close parenthesis to the power 6")[$attach(( x^(15) ), t: 6)$]

#solutionbox[
#emph[x]#super[90]
]

#math.equation(block: true, alt: "open parenthesis y to the power 12 close parenthesis to the power 8")[$attach(( y^(12) ), t: 8)$]

#math.equation(block: true, alt: "open parenthesis x squared close parenthesis to the power y")[$attach(( x^(2) ), t: y)$]

#solutionbox[
#emph[x]#super[2#emph[y]]
]

#math.equation(block: true, alt: "open parenthesis y cubed close parenthesis to the power x")[$attach(( y^(3) ), t: x)$]

#math.equation(block: true, alt: "open parenthesis 5 to the power x close parenthesis to the power y")[$attach(( 5^(x) ), t: y)$]

#solutionbox[
5#super[#emph[x]#emph[y]]
]

#math.equation(block: true, alt: "open parenthesis 7 to the power a close parenthesis to the power b")[$attach(( 7^(a) ), t: b)$]

#strong[Simplify Expressions Using the Product to a Power Property]

In the following exercises, simplify each expression using the Product to a Power Property.

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

#solutionbox[
25#emph[a]#super[2]
]

#math.equation(block: true, alt: "open parenthesis 7 x close parenthesis squared")[$attach(( 7 x ), t: 2)$]

#math.equation(block: true, alt: "open parenthesis minus 6 m close parenthesis cubed")[$attach(( − 6 m ), t: 3)$]

#solutionbox[
−216#emph[m]#super[3]
]

#math.equation(block: true, alt: "open parenthesis minus 9 n close parenthesis cubed")[$attach(( − 9 n ), t: 3)$]

#math.equation(block: true, alt: "open parenthesis 4 r s close parenthesis squared")[$attach(( 4 r s ), t: 2)$]

#solutionbox[
16#emph[r]#super[2]#emph[s]#super[2]
]

#math.equation(block: true, alt: "open parenthesis 5 a b close parenthesis cubed")[$attach(( 5 a b ), t: 3)$]

#math.equation(block: true, alt: "open parenthesis 4 x y z close parenthesis to the power 4")[$attach(( 4 x y z ), t: 4)$]

#solutionbox[
256#emph[x]#super[4]#emph[y]#super[4]#emph[z]#super[4]
]

#math.equation(block: true, alt: "open parenthesis minus 5 a b c close parenthesis cubed")[$attach(( − 5 a b c ), t: 3)$]

#strong[Simplify Expressions by Applying Several Properties]

In the following exercises, simplify each expression.

#math.equation(block: true, alt: "open parenthesis x squared close parenthesis to the power 4 times open parenthesis x cubed close parenthesis squared")[$attach(( x^(2) ), t: 4) · attach(( x^(3) ), t: 2)$]

#solutionbox[
#emph[x]#super[14]
]

#math.equation(block: true, alt: "open parenthesis y to the power 4 close parenthesis cubed times open parenthesis y to the power 5 close parenthesis squared")[$attach(( y^(4) ), t: 3) · attach(( y^(5) ), t: 2)$]

#math.equation(block: true, alt: "open parenthesis a squared close parenthesis to the power 6 times open parenthesis a cubed close parenthesis to the power 8")[$attach(( a^(2) ), t: 6) · attach(( a^(3) ), t: 8)$]

#solutionbox[
#emph[a]#super[36]
]

#math.equation(block: true, alt: "open parenthesis b to the power 7 close parenthesis to the power 5 times open parenthesis b squared close parenthesis to the power 6")[$attach(( b^(7) ), t: 5) · attach(( b^(2) ), t: 6)$]

#math.equation(block: true, alt: "open parenthesis 3 x close parenthesis squared open parenthesis 5 x close parenthesis")[$attach(( 3 x ), t: 2) ( 5 x )$]

#solutionbox[
45#emph[x]#super[3]
]

#math.equation(block: true, alt: "open parenthesis 2 y close parenthesis cubed open parenthesis 6 y close parenthesis")[$attach(( 2 y ), t: 3) ( 6 y )$]

#math.equation(block: true, alt: "open parenthesis 5 a close parenthesis squared open parenthesis 2 a close parenthesis cubed")[$attach(( 5 a ), t: 2) attach(( 2 a ), t: 3)$]

#solutionbox[
200#emph[a]#super[5]
]

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

#math.equation(block: true, alt: "open parenthesis 2 m to the power 6 close parenthesis cubed")[$attach(( 2 m^(6) ), t: 3)$]

#solutionbox[
8#emph[m]#super[18]
]

#math.equation(block: true, alt: "open parenthesis 3 y squared close parenthesis to the power 4")[$attach(( 3 y^(2) ), t: 4)$]

#math.equation(block: true, alt: "open parenthesis 10 x squared y close parenthesis cubed")[$attach(( 10 x^(2) y ), t: 3)$]

#solutionbox[
1,000#emph[x]#super[6]#emph[y]#super[3]
]

#math.equation(block: true, alt: "open parenthesis 2 m n to the power 4 close parenthesis to the power 5")[$attach(( 2 m n^(4) ), t: 5)$]

#math.equation(block: true, alt: "open parenthesis −2 a cubed b squared close parenthesis to the power 4")[$attach(( −2 a^(3) b^(2) ), t: 4)$]

#solutionbox[
16#emph[a]#super[12]#emph[b]#super[8]
]

#math.equation(block: true, alt: "open parenthesis −10 u squared v to the power 4 close parenthesis cubed")[$attach(( −10 u^(2) v^(4) ), t: 3)$]

#math.equation(block: true, alt: "open parenthesis the fraction 2 over 3 x squared y close parenthesis cubed")[$attach(( frac(2, 3) #h(0.1em) x^(2) y ), t: 3)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 8 over 27 x to the power 6 y cubed")[$frac(8, 27) #h(0.1em) x^(6) y^(3)$]
]

#math.equation(block: true, alt: "open parenthesis the fraction 7 over 9 p q to the power 4 close parenthesis squared")[$attach(( frac(7, 9) #h(0.1em) p q^(4) ), t: 2)$]

#math.equation(block: true, alt: "open parenthesis 8 a cubed close parenthesis squared open parenthesis 2 a close parenthesis to the power 4")[$attach(( 8 a^(3) ), t: 2) attach(( 2 a ), t: 4)$]

#solutionbox[
1,024#emph[a]#super[10]
]

#math.equation(block: true, alt: "open parenthesis 5 r squared close parenthesis cubed open parenthesis 3 r close parenthesis squared")[$attach(( 5 r^(2) ), t: 3) attach(( 3 r ), t: 2)$]

#math.equation(block: true, alt: "open parenthesis 10 p to the power 4 close parenthesis cubed open parenthesis 5 p to the power 6 close parenthesis squared")[$attach(( 10 p^(4) ), t: 3) attach(( 5 p^(6) ), t: 2)$]

#solutionbox[
25,000#emph[p]#super[24]
]

#math.equation(block: true, alt: "open parenthesis 4 x cubed close parenthesis cubed open parenthesis 2 x to the power 5 close parenthesis to the power 4")[$attach(( 4 x^(3) ), t: 3) attach(( 2 x^(5) ), t: 4)$]

#math.equation(block: true, alt: "open parenthesis the fraction 1 over 2 x squared y cubed close parenthesis to the power 4 open parenthesis 4 x to the power 5 y cubed close parenthesis squared")[$attach(( frac(1, 2) #h(0.1em) x^(2) y^(3) ), t: 4) attach(( 4 x^(5) y^(3) ), t: 2)$]

#solutionbox[
#emph[x]#super[18]#emph[y]#super[18]
]

#math.equation(block: true, alt: "open parenthesis the fraction 1 over 3 m cubed n squared close parenthesis to the power 4 open parenthesis 9 m to the power 8 n cubed close parenthesis squared")[$attach(( frac(1, 3) #h(0.1em) m^(3) n^(2) ), t: 4) attach(( 9 m^(8) n^(3) ), t: 2)$]

#math.equation(block: true, alt: "open parenthesis 3 m squared n close parenthesis squared open parenthesis 2 m n to the power 5 close parenthesis to the power 4")[$attach(( 3 m^(2) n ), t: 2) attach(( 2 m n^(5) ), t: 4)$]

#solutionbox[
144#emph[m]#super[8]#emph[n]#super[22]
]

#math.equation(block: true, alt: "open parenthesis 2 p q to the power 4 close parenthesis cubed open parenthesis 5 p to the power 6 q close parenthesis squared")[$attach(( 2 p q^(4) ), t: 3) attach(( 5 p^(6) q ), t: 2)$]

#strong[Multiply Monomials]

In the following exercises, multiply the following monomials.

#math.equation(block: true, alt: "open parenthesis 12 x squared close parenthesis open parenthesis −5 x to the power 4 close parenthesis")[$( 12 x^(2) ) ( −5 x^(4) )$]

#solutionbox[
−60#emph[x]#super[6]
]

#math.equation(block: true, alt: "open parenthesis −10 y cubed close parenthesis open parenthesis 7 y squared close parenthesis")[$( −10 y^(3) ) ( 7 y^(2) )$]

#math.equation(block: true, alt: "open parenthesis −8 u to the power 6 close parenthesis open parenthesis −9 u close parenthesis")[$( −8 u^(6) ) ( −9 u )$]

#solutionbox[
72#emph[u]#super[7]
]

#math.equation(block: true, alt: "open parenthesis −6 c to the power 4 close parenthesis open parenthesis −12 c close parenthesis")[$( −6 c^(4) ) ( −12 c )$]

#math.equation(block: true, alt: "open parenthesis the fraction 1 over 5 r to the power 8 close parenthesis open parenthesis 20 r cubed close parenthesis")[$( frac(1, 5) #h(0.1em) r^(8) ) ( 20 r^(3) )$]

#solutionbox[
4#emph[r]#super[11]
]

#math.equation(block: true, alt: "open parenthesis the fraction 1 over 4 a to the power 5 close parenthesis open parenthesis 36 a squared close parenthesis")[$( frac(1, 4) #h(0.1em) a^(5) ) ( 36 a^(2) )$]

#math.equation(block: true, alt: "open parenthesis 4 a cubed b close parenthesis open parenthesis 9 a squared b to the power 6 close parenthesis")[$( 4 a^(3) b ) ( 9 a^(2) b^(6) )$]

#solutionbox[
36#emph[a]#super[5]#emph[b]#super[7]
]

#math.equation(block: true, alt: "open parenthesis 6 m to the power 4 n cubed close parenthesis open parenthesis 7 m n to the power 5 close parenthesis")[$( 6 m^(4) n^(3) ) ( 7 m n^(5) )$]

#math.equation(block: true, alt: "open parenthesis the fraction 4 over 7 x y squared close parenthesis open parenthesis 14 x y cubed close parenthesis")[$( frac(4, 7) #h(0.1em) x y^(2) ) ( 14 x y^(3) )$]

#solutionbox[
8#emph[x]#super[2]#emph[y]#super[5]
]

#math.equation(block: true, alt: "open parenthesis the fraction 5 over 8 u cubed v close parenthesis open parenthesis 24 u to the power 5 v close parenthesis")[$( frac(5, 8) #h(0.1em) u^(3) v ) ( 24 u^(5) v )$]

#math.equation(block: true, alt: "open parenthesis the fraction 2 over 3 x squared y close parenthesis open parenthesis the fraction 3 over 4 x y squared close parenthesis")[$( frac(2, 3) #h(0.1em) x^(2) y ) ( frac(3, 4) #h(0.1em) x y^(2) )$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 2 x cubed y cubed")[$frac(1, 2) #h(0.1em) x^(3) y^(3)$]
]

#math.equation(block: true, alt: "open parenthesis the fraction 3 over 5 m cubed n squared close parenthesis open parenthesis the fraction 5 over 9 m squared n cubed close parenthesis")[$( frac(3, 5) #h(0.1em) m^(3) n^(2) ) ( frac(5, 9) #h(0.1em) m^(2) n^(3) )$]


==== Everyday Math

#strong[Email] Janet emails a joke to six of her friends and tells them to forward it to six of their friends, who forward it to six of their friends, and so on. The number of people who receive the email on the second round is #math.equation(block: false, alt: "6 squared ,")[$6^(2) ,$] on the third round is #math.equation(block: false, alt: "6 cubed ,")[$6^(3) ,$] as shown in the table. How many people will receive the email on the eighth round? Simplify the expression to show the number of people who receive the email.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Round], [Number of people]),
  [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "6")[$6$]],
  [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "6 squared")[$6^(2)$]],
  [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "6 cubed")[$6^(3)$]],
  [#math.equation(block: false, alt: "and so on")[$…$]], [#math.equation(block: false, alt: "and so on")[$…$]],
  [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "?")[$?$]],
))

#solutionbox[
1,679,616
]

#strong[Salary] Raul’s boss gives him a #math.equation(block: false, alt: "5%")[$"5%"$] raise every year on his birthday. This means that each year, Raul’s salary is #math.equation(block: false, alt: "1.05")[$1.05$] times his last year’s salary. If his original salary was #math.equation(block: false, alt: "$40,000")[$"$40,000"$], his salary after #math.equation(block: false, alt: "1")[$1$] year was #math.equation(block: false, alt: "$40,000 open parenthesis 1.05 close parenthesis ,")[$"$40,000" ( 1.05 ) ,$] after #math.equation(block: false, alt: "2")[$2$] years was #math.equation(block: false, alt: "$40,000 open parenthesis 1.05 close parenthesis squared ,")[$"$40,000" attach(( 1.05 ), t: 2) ,$] after #math.equation(block: false, alt: "3")[$3$] years was #math.equation(block: false, alt: "$40,000 open parenthesis 1.05 close parenthesis cubed ,")[$"$40,000" attach(( 1.05 ), t: 3) ,$] as shown in the table below. What will Raul’s salary be after #math.equation(block: false, alt: "10")[$10$] years? Simplify the expression, to show Raul’s salary in dollars.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Year], [Salary]),
  [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "$40,000 open parenthesis 1.05 close parenthesis")[$"$40,000" ( 1.05 )$]],
  [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "$40,000 open parenthesis 1.05 close parenthesis squared")[$"$40,000" attach(( 1.05 ), t: 2)$]],
  [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "$40,000 open parenthesis 1.05 close parenthesis cubed")[$"$40,000" attach(( 1.05 ), t: 3)$]],
  [#math.equation(block: false, alt: "and so on")[$…$]], [#math.equation(block: false, alt: "and so on")[$…$]],
  [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "?")[$?$]],
))


==== Writing Exercises

Use the Product Property for Exponents to explain why #math.equation(block: false, alt: "x times x equals x squared .")[$x · x = x^(2) .$]

#solutionbox[
Answers will vary.
]

Explain why #math.equation(block: false, alt: "−5 cubed equals open parenthesis −5 close parenthesis cubed")[$−5^(3) = attach(( −5 ), t: 3)$] but #math.equation(block: false, alt: "−5 to the power 4 not equal to open parenthesis −5 close parenthesis to the power 4 .")[$−5^(4) ≠ attach(( −5 ), t: 4) .$]

Jorge thinks #math.equation(block: false, alt: "open parenthesis the fraction 1 over 2 close parenthesis squared")[$attach(( frac(1, 2) ), t: 2)$] is #math.equation(block: false, alt: "1 .")[$1 .$] What is wrong with his reasoning?

#solutionbox[
Answers will vary.
]

Explain why #math.equation(block: false, alt: "x cubed times x to the power 5")[$x^(3) · x^(5)$] is #math.equation(block: false, alt: "x to the power 8 ,")[$x^(8) ,$] and not #math.equation(block: false, alt: "x to the power 15 .")[$x^(15) .$]


==== Self Check

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

#figure(figph[A self-assessment checklist for students to rate their understanding of simplifying expressions with exponents and multiplying monomials, using categories like 'Confidently' and 'No-I don't get it!'], alt: "A self-assessment checklist for students to rate their understanding of simplifying expressions with exponents and multiplying monomials, using categories like 'Confidently' and 'No-I don't get it!'", caption: none)

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?