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== 9.1#h(0.6em)Simplify and Use Square Roots

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

Simplify: ⓐ #math.equation(block: false, alt: "9 squared")[$9^(2)$] ⓑ #math.equation(block: false, alt: "open parenthesis −9 close parenthesis squared")[$attach(( −9 ), t: 2)$] ⓒ #math.equation(block: false, alt: "− 9 squared")[$"−" 9^(2)$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
ⓐ #math.equation(block: false, alt: "81")[$81$] ⓑ #math.equation(block: false, alt: "81")[$81$] ⓒ #math.equation(block: false, alt: "minus 81")[$− 81$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Round 3.846 to the nearest hundredth. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "3.85")[$3.85$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For each number, identify whether it is a real number or not a real number: #linebreak() 
ⓐ #math.equation(block: false, alt: "− the square root of 100")[$"−" sqrt(100)$] ⓑ #math.equation(block: false, alt: "the square root of −100")[$sqrt(−100)$]. #linebreak() 
If you missed this problem, review .

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


=== Simplify Expressions with Square Roots

Remember that when a number #math.equation(block: false, alt: "n")[$n$] is multiplied by itself, we write #math.equation(block: false, alt: "n squared")[$n^(2)$] and read it “n squared.” For example, #math.equation(block: false, alt: "15 squared")[$15^(2)$] reads as “15 squared,” and 225 is called the square of 15, since #math.equation(block: false, alt: "15 squared equals 225")[$15^(2) = 225$].

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

If #math.equation(block: false, alt: "n squared equals m")[$n^(2) = m$], then #math.equation(block: false, alt: "m")[$m$] is the square of #math.equation(block: false, alt: "n")[$n$].
]

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because 225 is the square of 15, we can also say that 15 is a square root of 225. A number whose square is #math.equation(block: false, alt: "m")[$m$] is called a #emph[square root] of #math.equation(block: false, alt: "m")[$m$].

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

If #math.equation(block: false, alt: "n squared equals m")[$n^(2) = m$], then #math.equation(block: false, alt: "n")[$n$] is a square root of #math.equation(block: false, alt: "m")[$m$].
]

Notice #math.equation(block: false, alt: "open parenthesis −15 close parenthesis squared equals 225")[$attach(( −15 ), t: 2) = 225$] also, so #math.equation(block: false, alt: "−15")[$−15$] is also a square root of 225. Therefore, both 15 and #math.equation(block: false, alt: "−15")[$−15$] are square roots of 225.

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The #emph[radical sign,] #math.equation(block: false, alt: "the square root of m")[$sqrt(m)$], denotes the positive square root. The positive square root is also called the principal square root.

We also use the radical sign for the square root of zero. Because #math.equation(block: false, alt: "0 squared equals 0")[$0^(2) = 0$], #math.equation(block: false, alt: "the square root of 0 equals 0")[$sqrt(0) = 0$]. Notice that zero has only one square root.

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

#figure(figph[This figure is a picture of an m inside a square root sign. The sign is labeled as a radical sign and the m is labeled as the radicand.], alt: "This figure is a picture of an m inside a square root sign. The sign is labeled as a radical sign and the m is labeled as the radicand.", caption: none)

#math.equation(block: false, alt: "the square root of m")[$sqrt(m)$] is read as “the square root of #math.equation(block: false, alt: "m")[$m$].”

If #math.equation(block: false, alt: "m equals n squared")[$m = n^(2)$], then #math.equation(block: false, alt: "the square root of m equals n")[$sqrt(m) = n$], for #math.equation(block: false, alt: "n greater than or equal to 0")[$n ≥ 0$].

The square root of #math.equation(block: false, alt: "m")[$m$], #math.equation(block: false, alt: "the square root of m")[$sqrt(m)$], is the non-negative number whose square is #math.equation(block: false, alt: "m")[$m$].
]

Since 15 is the positive square root of 225, we write #math.equation(block: false, alt: "the square root of 225 equals 15")[$sqrt(225) = 15$]. Fill in  to make a table of square roots you can refer to as you work this chapter.

#figure(figph[This table has fifteen columns and two rows. The first row contains the following numbers: the square root of 1, the square root of 4, the square root of 9, the square root of 16, the square root of 25, the square root of 36, the square root of 49, the square root of 64, the square root of 81, the square root of 100, the square root of 121, the square root of 144, the square root of 169, the square root of 196, and the square root of 225. The second row is completely empty except for the last column. The number 15 is in the last column.], alt: "This table has fifteen columns and two rows. The first row contains the following numbers: the square root of 1, the square root of 4, the square root of 9, the square root of 16, the square root of 25, the square root of 36, the square root of 49, the square root of 64, the square root of 81, the square root of 100, the square root of 121, the square root of 144, the square root of 169, the square root of 196, and the square root of 225. The second row is completely empty except for the last column. The number 15 is in the last column.", caption: none)

We know that every positive number has two square roots and the radical sign indicates the positive one. We write #math.equation(block: false, alt: "the square root of 225 equals 15")[$sqrt(225) = 15$]. If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, #math.equation(block: false, alt: "− the square root of 225 equals −15")[$"−" sqrt(225) = −15$].

#examplebox("Example 1")[][
Simplify: ⓐ #math.equation(block: false, alt: "the square root of 36")[$sqrt(36)$] ⓑ #math.equation(block: false, alt: "the square root of 196")[$sqrt(196)$] ⓒ #math.equation(block: false, alt: "− the square root of 81")[$"−" sqrt(81)$] ⓓ #math.equation(block: false, alt: "− the square root of 289")[$"−" sqrt(289)$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 36")[$sqrt(36)$]]),
  [Since #math.equation(block: false, alt: "6 squared equals 36")[$6^(2) = 36$]], [6],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 196")[$sqrt(196)$]]),
  [Since #math.equation(block: false, alt: "14 squared equals 196")[$14^(2) = 196$]], [14],
))

ⓒ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "− the square root of 81")[$"−" sqrt(81)$]]),
  [The negative is in front of the radical sign.], [−9],
))

ⓓ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "− the square root of 289")[$"−" sqrt(289)$]]),
  [The negative is in front of the radical sign.], [−17],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "− the square root of 49")[$"−" sqrt(49)$] ⓑ #math.equation(block: false, alt: "the square root of 225")[$sqrt(225)$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "−7")[$−7$] ⓑ #math.equation(block: false, alt: "15")[$15$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "the square root of 64")[$sqrt(64)$] ⓑ #math.equation(block: false, alt: "− the square root of 121")[$"−" sqrt(121)$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "8")[$8$] ⓑ #math.equation(block: false, alt: "−11")[$−11$]
]
]

#examplebox("Example 2")[][
Simplify: ⓐ #math.equation(block: false, alt: "the square root of −169")[$sqrt(−169)$] ⓑ #math.equation(block: false, alt: "− the square root of 64")[$"−" sqrt(64)$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of −169")[$sqrt(−169)$]]),
  [There is no real number whose square is −169.], [#math.equation(block: false, alt: "the square root of −169")[$sqrt(−169)$] is not a real number.],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "− the square root of 64")[$"−" sqrt(64)$]]),
  [The negative is in front of the radical.], [−8],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "the square root of −196")[$sqrt(−196)$] ⓑ #math.equation(block: false, alt: "− the square root of 81")[$"−" sqrt(81)$].

#solutionbox[
ⓐ not a real number ⓑ #math.equation(block: false, alt: "−9")[$−9$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "− the square root of 49")[$"−" sqrt(49)$] ⓑ #math.equation(block: false, alt: "the square root of −121")[$sqrt(−121)$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "−7")[$−7$] ⓑ not a real number
]
]

When using the order of operations to simplify an expression that has square roots, we treat the radical as a grouping symbol.

#examplebox("Example 3")[][
Simplify: ⓐ #math.equation(block: false, alt: "the square root of 25 plus the square root of 144")[$sqrt(25) + sqrt(144)$] ⓑ #math.equation(block: false, alt: "the square root of 25 plus 144")[$sqrt(25 + 144)$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 25 plus the square root of 144")[$sqrt(25) + sqrt(144)$]]),
  [Use the order of operations.], [#math.equation(block: false, alt: "5 plus 12")[$5 + 12$]],
  [Simplify.], [17],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  [], [#math.equation(block: false, alt: "the square root of 25 plus 144")[$sqrt(25 + 144)$]],
  [Simplify under the radical sign.], [#math.equation(block: false, alt: "the square root of 169")[$sqrt(169)$]],
  [Simplify.], [13],
  [Notice the different answers in parts ⓐ and ⓑ !], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "the square root of 9 plus the square root of 16")[$sqrt(9) + sqrt(16)$] ⓑ #math.equation(block: false, alt: "the square root of 9 plus 16")[$sqrt(9 + 16)$].

#solutionbox[
ⓐ 7 ⓑ 5
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "the square root of 64 plus 225")[$sqrt(64 + 225)$] ⓑ #math.equation(block: false, alt: "the square root of 64 plus the square root of 225")[$sqrt(64) + sqrt(225)$].

#solutionbox[
ⓐ 17 ⓑ #math.equation(block: false, alt: "23")[$23$]
]
]


=== Estimate Square Roots

So far we have only considered square roots of perfect square numbers. The square roots of other numbers are not whole numbers. Look at  below.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Number], [Square Root]),
  [4], [#math.equation(block: false, alt: "the square root of 4")[$sqrt(4)$]
= 2],
  [5], [#math.equation(block: false, alt: "the square root of 5")[$sqrt(5)$]],
  [6], [#math.equation(block: false, alt: "the square root of 6")[$sqrt(6)$]],
  [7], [#math.equation(block: false, alt: "the square root of 7")[$sqrt(7)$]],
  [8], [#math.equation(block: false, alt: "the square root of 8")[$sqrt(8)$]],
  [9], [#math.equation(block: false, alt: "the square root of 9")[$sqrt(9)$] = 3],
))

The square roots of numbers between 4 and 9 must be between the two consecutive whole numbers 2 and 3, and they are not whole numbers. Based on the pattern in the table above, we could say that #math.equation(block: false, alt: "the square root of 5")[$sqrt(5)$] must be between 2 and 3. Using inequality symbols, we write:

#math.equation(block: true, alt: "2 less than the square root of 5 less than 3")[$2 < sqrt(5) < 3$]#examplebox("Example 4")[][
Estimate #math.equation(block: false, alt: "the square root of 60")[$sqrt(60)$] between two consecutive whole numbers.

#solutionbox[
Think of the perfect square numbers closest to 60. Make a small table of these perfect squares and their squares roots.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#figure(figph[A table lists four perfect squares: thirty-six, forty-nine, sixty-four, and eighty-one. The corresponding square roots are listed: six, seven, eight, and nine. Arrows indicate that the value of square root of sixty lies between seven and eight.], alt: "A table lists four perfect squares: thirty-six, forty-nine, sixty-four, and eighty-one. The corresponding square roots are listed: six, seven, eight, and nine. Arrows indicate that the value of square root of sixty lies between seven and eight.", caption: none)], []),
  [Locate 60 between two consecutive perfect squares.], [#figure(figph[A mathematical inequality displaying the relationship 49 \< 60 \< 64, with the number 60 highlighted in red to emphasize its position between 49 and 64.], alt: "A mathematical inequality displaying the relationship 49 < 60 < 64, with the number 60 highlighted in red to emphasize its position between 49 and 64.", caption: none)],
  [#math.equation(block: false, alt: "the square root of 60")[$sqrt(60)$] is between their square roots.], [#figure(figph[A mathematical inequality showing that 7 is less than the square root of 60, which is less than 8. The number 60 is highlighted in red beneath the square root symbol.], alt: "A mathematical inequality showing that 7 is less than the square root of 60, which is less than 8. The number 60 is highlighted in red beneath the square root symbol.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Estimate the square root #math.equation(block: false, alt: "the square root of 38")[$sqrt(38)$] between two consecutive whole numbers.

#solutionbox[
#math.equation(block: true, alt: "6 less than the square root of 38 less than 7")[$6 < sqrt(38) < 7$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Estimate the square root #math.equation(block: false, alt: "the square root of 84")[$sqrt(84)$] between two consecutive whole numbers.

#solutionbox[
#math.equation(block: true, alt: "9 less than the square root of 84 less than 10")[$9 < sqrt(84) < 10$]
]
]


=== Approximate Square Roots

There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find them. Find the #math.equation(block: false, alt: "the square root of x")[$sqrt(x)$] key on your calculator. You will use this key to approximate square roots.

When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact square root. It is an approximation, accurate to the number of digits shown on your calculator’s display. The symbol for an approximation is #math.equation(block: false, alt: "approximately equals")[$≈$] and it is read ‘approximately.’

Suppose your calculator has a 10-digit display. You would see that

#math.equation(block: true, alt: "the square root of 5 approximately equals 2.236067978")[$#h(3.4em) sqrt(5) ≈ 2.236067978$]If we wanted to round #math.equation(block: false, alt: "the square root of 5")[$sqrt(5)$] to two decimal places, we would say

#math.equation(block: true, alt: "the square root of 5 approximately equals 2.24")[$sqrt(5) ≈ 2.24$]How do we know these values are approximations and not the exact values? Look at what happens when we square them:

#math.equation(block: true, alt: "open parenthesis 2.236067978 close parenthesis squared, equals, 5.000000002; open parenthesis 2.24 close parenthesis squared, equals, 5.0176")[$attach(( 2.236067978 ), t: 2) & = & 5.000000002 \ attach(( 2.24 ), t: 2) & = & 5.0176$]Their squares are close to 5, but are not exactly equal to 5.

Using the square root key on a calculator and then rounding to two decimal places, we can find:

#math.equation(block: true, alt: "the square root of 4, equals, 2; the square root of 5, approximately equals, 2.24; the square root of 6, approximately equals, 2.45; the square root of 7, approximately equals, 2.65; the square root of 8, approximately equals, 2.83; the square root of 9, equals, 3")[$sqrt(4) & = & 2 \ sqrt(5) & ≈ & 2.24 \ sqrt(6) & ≈ & 2.45 \ sqrt(7) & ≈ & 2.65 \ sqrt(8) & ≈ & 2.83 \ sqrt(9) & = & 3$]#examplebox("Example 5")[][
Round #math.equation(block: false, alt: "the square root of 17")[$sqrt(17)$] to two decimal places.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 17")[$sqrt(17)$]]),
  [Use the calculator square root key.], [4.123105626...],
  [Round to two decimal places.], [4.12],
  [], [#math.equation(block: false, alt: "the square root of 17 approximately equals 4.12")[$sqrt(17) ≈ 4.12$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Round #math.equation(block: false, alt: "the square root of 11")[$sqrt(11)$] to two decimal places.

#solutionbox[
#math.equation(block: true, alt: "approximately equals 3.32")[$≈ 3.32$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Round #math.equation(block: false, alt: "the square root of 13")[$sqrt(13)$] to two decimal places.

#solutionbox[
#math.equation(block: true, alt: "approximately equals 3.61")[$≈ 3.61$]
]
]


=== Simplify Variable Expressions with Square Roots

What if we have to find a square root of an expression with a variable? Consider #math.equation(block: false, alt: "the square root of 9 x squared")[$sqrt(9 x^(2))$]. Can you think of an expression whose square is #math.equation(block: false, alt: "9 x squared")[$9 x^(2)$]?

#math.equation(block: true, alt: "open parenthesis ? close parenthesis squared, equals, 9 x squared; open parenthesis 3 x close parenthesis squared, equals, 9 x squared ,, so the square root of 9 x squared equals 3 x")[$attach(( ? ), t: 2) & = & 9 x^(2) &  &  &  \ attach(( 3 x ), t: 2) & = & 9 x^(2) , &  &  & "so" #h(0.2em) sqrt(9 x^(2)) = 3 x$]When we use the radical sign to take the square root of a variable expression, we should specify that #math.equation(block: false, alt: "x greater than or equal to 0")[$x ≥ 0$] to make sure we get the #emph[principal square root].

However, in this chapter we will assume that each variable in a square-root expression represents a non-negative number and so we will not write #math.equation(block: false, alt: "x greater than or equal to 0")[$x ≥ 0$] next to every radical.

What about square roots of higher powers of variables? Think about the Power Property of Exponents we used in Chapter 6.

#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")[$attach(( a^(m) ), t: n) = a^(m · n)$]If we square #math.equation(block: false, alt: "a to the power m")[$a^(m)$], the exponent will become #math.equation(block: false, alt: "2 m")[$2 m$].

#math.equation(block: true, alt: "open parenthesis a to the power m close parenthesis squared equals a to the power 2 m")[$attach(( a^(m) ), t: 2) = a^(2 m)$]How does this help us take square roots? Let’s look at a few:

#math.equation(block: true, alt: "the square root of 25 u to the power 8 equals 5 u to the power 4, because open parenthesis 5 u to the power 4 close parenthesis squared equals 25 u to the power 8; the square root of 16 r to the power 20 equals 4 r to the power 10, because open parenthesis 4 r to the power 10 close parenthesis squared equals 16 r to the power 20; the square root of 196 q to the power 36 equals 14 q to the power 18, because open parenthesis 14 q to the power 18 close parenthesis squared equals 196 q to the power 36")[$sqrt(25 u^(8)) = 5 u^(4) & "because" #h(0.2em) attach(( 5 u^(4) ), t: 2) = 25 u^(8) \ sqrt(16 r^(20)) = 4 r^(10) & "because" #h(0.2em) attach(( 4 r^(10) ), t: 2) = 16 r^(20) \ sqrt(196 q^(36)) = 14 q^(18) & "because" #h(0.2em) attach(( 14 q^(18) ), t: 2) = 196 q^(36)$]#examplebox("Example 6")[][
Simplify: ⓐ #math.equation(block: false, alt: "the square root of x to the power 6")[$sqrt(x^(6))$] ⓑ #math.equation(block: false, alt: "the square root of y to the power 16")[$sqrt(y^(16))$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of x to the power 6")[$sqrt(x^(6))$]]),
  [#math.equation(block: false, alt: "Since open parenthesis x cubed close parenthesis squared equals x to the power 6 .")[$"Since" #h(0.2em) attach(( x^(3) ), t: 2) = x^(6) .$]], [#math.equation(block: false, alt: "x cubed")[$x^(3)$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of y to the power 16")[$sqrt(y^(16))$]]),
  [#math.equation(block: false, alt: "Since open parenthesis y to the power 8 close parenthesis squared equals y to the power 16 .")[$"Since" #h(0.2em) attach(( y^(8) ), t: 2) = y^(16) .$]], [#math.equation(block: false, alt: "y to the power 8")[$y^(8)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "the square root of y to the power 8")[$sqrt(y^(8))$] ⓑ #math.equation(block: false, alt: "the square root of z to the power 12")[$sqrt(z^(12))$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "the square root of m to the power 4")[$sqrt(m^(4))$] ⓑ #math.equation(block: false, alt: "the square root of b to the power 10")[$sqrt(b^(10))$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "m squared")[$m^(2)$] ⓑ #math.equation(block: false, alt: "b to the power 5")[$b^(5)$]
]
]

#examplebox("Example 7")[][
Simplify: #math.equation(block: false, alt: "the square root of 16 n squared")[$sqrt(16 n^(2))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 16 n squared")[$sqrt(16 n^(2))$]]),
  [#math.equation(block: false, alt: "Since open parenthesis 4 n close parenthesis squared equals 16 n squared .")[$"Since" #h(0.2em) attach(( 4 n ), t: 2) = 16 n^(2) .$]], [#math.equation(block: false, alt: "4 n")[$4 n$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 64 x squared")[$sqrt(64 x^(2))$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 169 y squared")[$sqrt(169 y^(2))$].

#solutionbox[
#math.equation(block: true, alt: "13 y")[$13 y$]
]
]

#examplebox("Example 8")[][
Simplify: #math.equation(block: false, alt: "− the square root of 81 c squared")[$"−" sqrt(81 c^(2))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "− the square root of 81 c squared")[$"−" sqrt(81 c^(2))$]]),
  [#math.equation(block: false, alt: "Since open parenthesis 9 c close parenthesis squared equals 81 c squared .")[$"Since" #h(0.2em) attach(( 9 c ), t: 2) = 81 c^(2) .$]], [#math.equation(block: false, alt: "−9 c")[$−9 c$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "− the square root of 121 y squared")[$"−" sqrt(121 y^(2))$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "− the square root of 100 p squared")[$"−" sqrt(100 p^(2))$].

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

#examplebox("Example 9")[][
Simplify: #math.equation(block: false, alt: "the square root of 36 x squared y squared")[$sqrt(36 x^(2) y^(2))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 36 x squared y squared")[$sqrt(36 x^(2) y^(2))$]]),
  [#math.equation(block: false, alt: "Since open parenthesis 6 x y close parenthesis squared equals 36 x squared y squared .")[$"Since" #h(0.2em) attach(( 6 x y ), t: 2) = 36 x^(2) y^(2) .$]], [#math.equation(block: false, alt: "6 x y")[$6 x y$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 100 a squared b squared")[$sqrt(100 a^(2) b^(2))$].

#solutionbox[
#math.equation(block: true, alt: "10 a b")[$10 a b$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 225 m squared n squared")[$sqrt(225 m^(2) n^(2))$].

#solutionbox[
#math.equation(block: true, alt: "15 m n")[$15 m n$]
]
]

#examplebox("Example 10")[][
Simplify: #math.equation(block: false, alt: "the square root of 64 p to the power 64")[$sqrt(64 p^(64))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 64 p to the power 64")[$sqrt(64 p^(64))$]]),
  [#math.equation(block: false, alt: "Since open parenthesis 8 p to the power 32 close parenthesis squared equals 64 p to the power 64 .")[$"Since" #h(0.2em) attach(( 8 p^(32) ), t: 2) = 64 p^(64) .$]], [#math.equation(block: false, alt: "8 p to the power 32")[$8 p^(32)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 49 x to the power 30")[$sqrt(49 x^(30))$].

#solutionbox[
#math.equation(block: true, alt: "7 x to the power 15")[$7 x^(15)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 81 w to the power 36")[$sqrt(81 w^(36))$].

#solutionbox[
#math.equation(block: true, alt: "9 w to the power 18")[$9 w^(18)$]
]
]

#examplebox("Example 11")[][
Simplify: #math.equation(block: false, alt: "the square root of 121 a to the power 6 b to the power 8")[$sqrt(121 a^(6) b^(8))$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 121 a to the power 6 b to the power 8")[$sqrt(121 a^(6) b^(8))$]]),
  [#math.equation(block: false, alt: "Since open parenthesis 11 a cubed b to the power 4 close parenthesis squared equals 121 a to the power 6 b to the power 8 .")[$"Since" #h(0.2em) attach(( 11 a^(3) b^(4) ), t: 2) = 121 a^(6) b^(8) .$]], [#math.equation(block: false, alt: "11 a cubed b to the power 4")[$11 a^(3) b^(4)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 169 x to the power 10 y to the power 14")[$sqrt(169 x^(10) y^(14))$].

#solutionbox[
#math.equation(block: true, alt: "13 x to the power 5 y to the power 7")[$13 x^(5) y^(7)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 144 p to the power 12 q to the power 20")[$sqrt(144 p^(12) q^(20))$].

#solutionbox[
#math.equation(block: true, alt: "12 p to the power 6 q to the power 10")[$12 p^(6) q^(10)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Access this online resource for additional instruction and practice with square roots.

- #link("https://openstax.org/l/25SquareRoots")[Square Roots]
]


=== Key Concepts

- Note that the square root of a negative number is not a real number.
- Every positive number has two square roots, one positive and one negative. The positive square root of a positive number is the principal square root.
- We can estimate square roots using nearby perfect squares.
- We can approximate square roots using a calculator.
- When we use the radical sign to take the square root of a variable expression, we should specify that #math.equation(block: false, alt: "x greater than or equal to 0")[$x ≥ 0$] to make sure we get the principal square root.


==== Practice Makes Perfect

#strong[Simplify Expressions with Square Roots]

In the following exercises, simplify.

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

#solutionbox[
6
]

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

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

#solutionbox[
8
]

#math.equation(block: true, alt: "the square root of 169")[$sqrt(169)$]

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

#solutionbox[
3
]

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

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

#solutionbox[
10
]

#math.equation(block: true, alt: "the square root of 144")[$sqrt(144)$]

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

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

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

#math.equation(block: true, alt: "− the square root of 1")[$"−" sqrt(1)$]

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

#math.equation(block: true, alt: "− the square root of 121")[$"−" sqrt(121)$]

#math.equation(block: true, alt: "the square root of −121")[$sqrt(−121)$]

#solutionbox[
not a real number
]

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

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

#solutionbox[
not a real number
]

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

#math.equation(block: true, alt: "the square root of 9 plus 16")[$sqrt(9 + 16)$]

#solutionbox[
5
]

#math.equation(block: true, alt: "the square root of 25 plus 144")[$sqrt(25 + 144)$]

#math.equation(block: true, alt: "the square root of 9 plus the square root of 16")[$sqrt(9) + sqrt(16)$]

#solutionbox[
7
]

#math.equation(block: true, alt: "the square root of 25 plus the square root of 144")[$sqrt(25) + sqrt(144)$]

#strong[Estimate Square Roots]

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

#math.equation(block: true, alt: "the square root of 70")[$sqrt(70)$]

#solutionbox[
#math.equation(block: true, alt: "8 less than the square root of 70 less than 9")[$8 < sqrt(70) < 9$]
]

#math.equation(block: true, alt: "the square root of 55")[$sqrt(55)$]

#math.equation(block: true, alt: "the square root of 200")[$sqrt(200)$]

#solutionbox[
#math.equation(block: true, alt: "14 less than the square root of 200 less than 15")[$14 < sqrt(200) < 15$]
]

#math.equation(block: true, alt: "the square root of 172")[$sqrt(172)$]

#strong[Approximate Square Roots]

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

#math.equation(block: true, alt: "the square root of 19")[$sqrt(19)$]

#solutionbox[
4.36
]

#math.equation(block: true, alt: "the square root of 21")[$sqrt(21)$]

#math.equation(block: true, alt: "the square root of 53")[$sqrt(53)$]

#solutionbox[
7.28
]

#math.equation(block: true, alt: "the square root of 47")[$sqrt(47)$]

#strong[Simplify Variable Expressions with Square Roots]

In the following exercises, simplify.

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

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

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

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

#solutionbox[
#math.equation(block: true, alt: "a to the power 7")[$a^(7)$]
]

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

#math.equation(block: true, alt: "the square root of 49 x squared")[$sqrt(49 x^(2))$]

#solutionbox[
#math.equation(block: true, alt: "7 x")[$7 x$]
]

#math.equation(block: true, alt: "the square root of 100 y squared")[$sqrt(100 y^(2))$]

#math.equation(block: true, alt: "the square root of 121 m to the power 20")[$sqrt(121 m^(20))$]

#solutionbox[
#math.equation(block: true, alt: "11 m to the power 10")[$11 m^(10)$]
]

#math.equation(block: true, alt: "the square root of 25 h to the power 44")[$sqrt(25 h^(44))$]

#math.equation(block: true, alt: "the square root of 81 x to the power 36")[$sqrt(81 x^(36))$]

#solutionbox[
#math.equation(block: true, alt: "9 x to the power 18")[$9 x^(18)$]
]

#math.equation(block: true, alt: "the square root of 144 z to the power 84")[$sqrt(144 z^(84))$]

#math.equation(block: true, alt: "− the square root of 81 x to the power 18")[$"−" sqrt(81 x^(18))$]

#solutionbox[
#math.equation(block: true, alt: "−9 x to the power 9")[$−9 x^(9)$]
]

#math.equation(block: true, alt: "− the square root of 100 m to the power 32")[$"−" sqrt(100 m^(32))$]

#math.equation(block: true, alt: "− the square root of 64 a squared")[$"−" sqrt(64 a^(2))$]

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

#math.equation(block: true, alt: "− the square root of 25 x squared")[$"−" sqrt(25 x^(2))$]

#math.equation(block: true, alt: "the square root of 144 x squared y squared")[$sqrt(144 x^(2) y^(2))$]

#solutionbox[
#math.equation(block: true, alt: "12 x y")[$12 x y$]
]

#math.equation(block: true, alt: "the square root of 196 a squared b squared")[$sqrt(196 a^(2) b^(2))$]

#math.equation(block: true, alt: "the square root of 169 w to the power 8 y to the power 10")[$sqrt(169 w^(8) y^(10))$]

#solutionbox[
#math.equation(block: true, alt: "13 w to the power 4 y to the power 5")[$13 w^(4) y^(5)$]
]

#math.equation(block: true, alt: "the square root of 81 p to the power 24 q to the power 6")[$sqrt(81 p^(24) q^(6))$]

#math.equation(block: true, alt: "the square root of 9 c to the power 8 d to the power 12")[$sqrt(9 c^(8) d^(12))$]

#solutionbox[
#math.equation(block: true, alt: "3 c to the power 4 d to the power 6")[$3 c^(4) d^(6)$]
]

#math.equation(block: true, alt: "the square root of 36 r to the power 6 s to the power 20")[$sqrt(36 r^(6) s^(20))$]


==== Everyday Math

#strong[Decorating] Denise wants to have a square accent of designer tiles in her new shower. She can afford to buy 625 square centimeters of the designer tiles. How long can a side of the accent be?

#solutionbox[
25 centimeters
]

#strong[Decorating] Morris wants to have a square mosaic inlaid in his new patio. His budget allows for 2025 square inch tiles. How long can a side of the mosaic be?


==== Writing Exercises

Why is there no real number equal to #math.equation(block: false, alt: "the square root of −64")[$sqrt(−64)$]?

#solutionbox[
Answers will vary.
]

What is the difference between #math.equation(block: false, alt: "9 squared")[$9^(2)$] and #math.equation(block: false, alt: "the square root of 9")[$sqrt(9)$]?


==== Self Check

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

#figure(figph[This table has four columns and five rows. The columns are labeled, “I can…,” “Confidentally,” “With some help,” and “No – I don’t get it!” Under the “I can…,” column are, “simplify expressions with square roots.,” “estimate square roots.,” “approximate square roots.,” and “4) simplify variable expressions with square roots.” All the other rows under the different columns are empty.], alt: "This table has four columns and five rows. The columns are labeled, “I can…,” “Confidentally,” “With some help,” and “No – I don’t get it!” Under the “I can…,” column are, “simplify expressions with square roots.,” “estimate square roots.,” “approximate square roots.,” and “4) simplify variable expressions with square roots.” All the other rows under the different columns are empty.", caption: none)

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?