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== 9.2#h(0.6em)Simplify Square Roots

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

Simplify: #math.equation(block: false, alt: "the fraction 80 over 176")[$frac(80, 176)$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 over 11")[$frac(5, 11)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction n to the power 9 over n cubed")[$frac(n^(9), n^(3))$]. #linebreak() 
If you missed this problem, review .

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction q to the power 4 over q to the power 12")[$frac(q^(4), q^(12))$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over q to the power 8")[$frac(1, q^(8))$]
]
]

In the last section, we estimated the square root of a number between two consecutive whole numbers. We can say that #math.equation(block: false, alt: "the square root of 50")[$sqrt(50)$] is between 7 and 8. This is fairly easy to do when the numbers are small enough that we can use .

But what if we want to estimate #math.equation(block: false, alt: "the square root of 500")[$sqrt(500)$]? If we simplify the square root first, we’ll be able to estimate it easily. There are other reasons, too, to simplify square roots as you’ll see later in this chapter.

A square root is considered #emph[simplified] if its radicand contains no perfect square factors.

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

#math.equation(block: false, alt: "the square root of a")[$sqrt(a)$] is considered simplified if #math.equation(block: false, alt: "a")[$a$] has no perfect square factors.
]

So #math.equation(block: false, alt: "the square root of 31")[$sqrt(31)$] is simplified. But #math.equation(block: false, alt: "the square root of 32")[$sqrt(32)$] is not simplified, because 16 is a perfect square factor of 32.


=== Use the Product Property to Simplify Square Roots

The properties we will use to simplify expressions with square roots are similar to the properties of exponents. We know that #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")[$attach(( a b ), t: m) = a^(m) b^(m)$]. The corresponding property of square roots says that #math.equation(block: false, alt: "the square root of a b equals the square root of a times the square root of b")[$sqrt(a b) = sqrt(a) · sqrt(b)$].

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

If #emph[a], #emph[b] are non-negative real numbers, then #math.equation(block: false, alt: "the square root of a b equals the square root of a times the square root of b")[$sqrt(a b) = sqrt(a) · sqrt(b)$].
]

We use the Product Property of Square Roots to remove all perfect square factors from a radical. We will show how to do this in .

#examplebox("Example 1")[How To Use the Product Property to Simplify a Square Root][
Simplify: #math.equation(block: false, alt: "the square root of 50")[$sqrt(50)$].

#solutionbox[
#figure(figph[This figure has three columns and three rows. The first row says, “Step 1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect square factor.” It then says, “25 is the largest perfect square factor of 50. 50 equals 25 times 2. Always write the perfect square factor first.” Then it shows the square root of 50 and the square root of 25 times 2.], alt: "This figure has three columns and three rows. The first row says, “Step 1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect square factor.” It then says, “25 is the largest perfect square factor of 50. 50 equals 25 times 2. Always write the perfect square factor first.” Then it shows the square root of 50 and the square root of 25 times 2.", caption: none)

#figure(figph[The second row says, “Step 2. Use the product rule to rewrite the radical as the product of two radicals.” The second column is empty, but the third column shows the square root of 25 times the square root of 2.], alt: "The second row says, “Step 2. Use the product rule to rewrite the radical as the product of two radicals.” The second column is empty, but the third column shows the square root of 25 times the square root of 2.", caption: none)

#figure(figph[The third row says, “Step 3. Simplify the square root of the perfect square.” The second column is empty, but the third column shows 5 times the square root of 2.], alt: "The third row says, “Step 3. Simplify the square root of the perfect square.” The second column is empty, but the third column shows 5 times the square root of 2.", caption: none)
]
]

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

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 45")[$sqrt(45)$].

#solutionbox[
#math.equation(block: true, alt: "3 the square root of 5")[$3 sqrt(5)$]
]
]

Notice in the previous example that the simplified form of #math.equation(block: false, alt: "the square root of 50")[$sqrt(50)$] is #math.equation(block: false, alt: "5 the square root of 2")[$5 sqrt(2)$], which is the product of an integer and a square root. We always write the integer in front of the square root.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Simplify a square root using the product property.]

+ Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect-square factor.
+ Use the product rule to rewrite the radical as the product of two radicals.
+ Simplify the square root of the perfect square.
]

#examplebox("Example 2")[][
Simplify: #math.equation(block: false, alt: "the square root of 500")[$sqrt(500)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 500")[$sqrt(500)$]]),
  [Rewrite the radicand as a product using the #linebreak() largest perfect square factor.], [#math.equation(block: false, alt: "the square root of 100 times 5")[$sqrt(100 · 5)$]],
  [Rewrite the radical as the product of two radicals.], [#math.equation(block: false, alt: "the square root of 100 times the square root of 5")[$sqrt(100) · sqrt(5)$]],
  [Simplify.], [#math.equation(block: false, alt: "10 the square root of 5")[$10 sqrt(5)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 288")[$sqrt(288)$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 432")[$sqrt(432)$].

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

We could use the simplified form #math.equation(block: false, alt: "10 the square root of 5")[$10 sqrt(5)$] to estimate #math.equation(block: false, alt: "the square root of 500")[$sqrt(500)$]. We know #math.equation(block: false, alt: "the square root of 5")[$sqrt(5)$] is between 2 and 3, and #math.equation(block: false, alt: "the square root of 500")[$sqrt(500)$] is #math.equation(block: false, alt: "10 the square root of 5")[$10 sqrt(5)$]. So #math.equation(block: false, alt: "the square root of 500")[$sqrt(500)$] is between 20 and 30.

The next example is much like the previous examples, but with variables.

#examplebox("Example 3")[][
Simplify: #math.equation(block: false, alt: "the square root of x cubed")[$sqrt(x^(3))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of x cubed")[$sqrt(x^(3))$]]),
  [Rewrite the radicand as a product using the #linebreak() largest perfect square factor.], [#math.equation(block: false, alt: "the square root of x squared times x")[$sqrt(x^(2) · x)$]],
  [Rewrite the radical as the product of two radicals.], [#math.equation(block: false, alt: "the square root of x squared times the square root of x")[$sqrt(x^(2)) · sqrt(x)$]],
  [Simplify.], [#math.equation(block: false, alt: "x the square root of x")[$x sqrt(x)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of b to the power 5")[$sqrt(b^(5))$].

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

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

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

We follow the same procedure when there is a coefficient in the radical, too.

#examplebox("Example 4")[][
Simplify: #math.equation(block: false, alt: "the square root of 25 y to the power 5.")[$sqrt(25 y^(5) .)$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 25 y to the power 5")[$sqrt(25 y^(5))$]]),
  [Rewrite the radicand as a product using the #linebreak() largest perfect square factor.], [#math.equation(block: false, alt: "the square root of 25 y to the power 4 times y")[$sqrt(25 y^(4) · y)$]],
  [Rewrite the radical as the product of two radicals.], [#math.equation(block: false, alt: "the square root of 25 y to the power 4 times the square root of y")[$sqrt(25 y^(4)) · sqrt(y)$]],
  [Simplify.], [#math.equation(block: false, alt: "5 y squared the square root of y")[$5 y^(2) sqrt(y)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 16 x to the power 7")[$sqrt(16 x^(7))$].

#solutionbox[
#math.equation(block: true, alt: "4 x cubed the square root of x")[$4 x^(3) sqrt(x)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 49 v to the power 9")[$sqrt(49 v^(9))$].

#solutionbox[
#math.equation(block: true, alt: "7 v to the power 4 the square root of v")[$7 v^(4) sqrt(v)$]
]
]

In the next example both the constant and the variable have perfect square factors.

#examplebox("Example 5")[][
Simplify: #math.equation(block: false, alt: "the square root of 72 n to the power 7")[$sqrt(72 n^(7))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 72 n to the power 7")[$sqrt(72 n^(7))$]]),
  [Rewrite the radicand as a product using the #linebreak() largest perfect square factor.], [#math.equation(block: false, alt: "the square root of 36 n to the power 6 times 2 n")[$sqrt(36 n^(6) · 2 n)$]],
  [Rewrite the radical as the product of two radicals.], [#math.equation(block: false, alt: "the square root of 36 n to the power 6 times the square root of 2 n")[$sqrt(36 n^(6)) · sqrt(2 n)$]],
  [Simplify.], [#math.equation(block: false, alt: "6 n cubed the square root of 2 n")[$6 n^(3) sqrt(2 n)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 32 y to the power 5")[$sqrt(32 y^(5))$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 75 a to the power 9")[$sqrt(75 a^(9))$].

#solutionbox[
#math.equation(block: true, alt: "5 a to the power 4 the square root of 3 a")[$5 a^(4) sqrt(3 a)$]
]
]

#examplebox("Example 6")[][
Simplify: #math.equation(block: false, alt: "the square root of 63 u cubed v to the power 5")[$sqrt(63 u^(3) v^(5))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 63 u cubed v to the power 5")[$sqrt(63 u^(3) v^(5))$]]),
  [Rewrite the radicand as a product using the #linebreak() largest perfect square factor.], [#math.equation(block: false, alt: "the square root of 9 u squared v to the power 4 times 7 u v")[$sqrt(9 u^(2) v^(4) · 7 u v)$]],
  [Rewrite the radical as the product of two radicals.], [#math.equation(block: false, alt: "the square root of 9 u squared v to the power 4 times the square root of 7 u v")[$sqrt(9 u^(2) v^(4)) · sqrt(7 u v)$]],
  [Simplify.], [#math.equation(block: false, alt: "3 u v squared the square root of 7 u v")[$3 u v^(2) sqrt(7 u v)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 98 a to the power 7 b to the power 5")[$sqrt(98 a^(7) b^(5))$].

#solutionbox[
#math.equation(block: true, alt: "7 a cubed b squared the square root of 2 a b")[$7 a^(3) b^(2) attach(sqrt(2 a b))$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 180 m to the power 9 n to the power 11")[$sqrt(180 m^(9) n^(11))$].

#solutionbox[
#math.equation(block: true, alt: "6 m to the power 4 n to the power 5 the square root of 5 m n")[$6 m^(4) n^(5) sqrt(5 m n)$]
]
]

We have seen how to use the Order of Operations to simplify some expressions with radicals. To simplify #math.equation(block: false, alt: "the square root of 25 plus the square root of 144")[$sqrt(25) + sqrt(144)$] we must simplify each square root separately first, then add to get the sum of 17.

The expression #math.equation(block: false, alt: "the square root of 17 plus the square root of 7")[$sqrt(17) + sqrt(7)$] cannot be simplified—to begin we’d need to simplify each square root, but neither 17 nor 7 contains a perfect square factor.

In the next example, we have the sum of an integer and a square root. We simplify the square root but cannot add the resulting expression to the integer.

#examplebox("Example 7")[][
Simplify: #math.equation(block: false, alt: "3 plus the square root of 32")[$3 + sqrt(32)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "3 plus the square root of 32")[$3 + sqrt(32)$]]),
  [Rewrite the radicand as a product using the #linebreak() largest perfect square factor.], [#math.equation(block: false, alt: "3 plus the square root of 16 times 2")[$3 + sqrt(16 · 2)$]],
  [Rewrite the radical as the product of two radicals.], [#math.equation(block: false, alt: "3 plus the square root of 16 times the square root of 2")[$3 + sqrt(16) · sqrt(2)$]],
  [Simplify.], [#math.equation(block: false, alt: "3 plus 4 the square root of 2")[$3 + 4 sqrt(2)$]],
))

The terms are not like and so we cannot add them. Trying to add an integer and a radical is like trying to add an integer and a variable—they are not like terms!
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "5 plus the square root of 75")[$5 + sqrt(75)$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "2 plus the square root of 98")[$2 + sqrt(98)$].

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

The next example includes a fraction with a radical in the numerator. Remember that in order to simplify a fraction you need a common factor in the numerator and denominator.

#examplebox("Example 8")[][
Simplify: #math.equation(block: false, alt: "the fraction 4 minus the square root of 48 over 2")[$frac(4 − sqrt(48), 2)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 4 minus the square root of 48 over 2")[$frac(4 − sqrt(48), 2)$]]),
  [Rewrite the radicand as a product using the #linebreak() largest perfect square factor.], [#math.equation(block: false, alt: "the fraction 4 minus the square root of 16 times 3 over 2")[$frac(4 − sqrt(16 · 3), 2)$]],
  [Rewrite the radical as the product of two radicals.], [#math.equation(block: false, alt: "the fraction 4 minus the square root of 16 times the square root of 3 over 2")[$frac(4 − sqrt(16) · sqrt(3), 2)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 4 minus 4 the square root of 3 over 2")[$frac(4 − 4 sqrt(3), 2)$]],
  [Factor the common factor from the numerator.], [#math.equation(block: false, alt: "the fraction 4 open parenthesis 1 minus the square root of 3 close parenthesis over 2")[$frac(4 ( 1 − sqrt(3) ), 2)$]],
  [Remove the common factor, 2, from the #linebreak() numerator and denominator.], [#math.equation(block: false, alt: "the fraction 2 times 2 open parenthesis 1 minus the square root of 3 close parenthesis over 2")[$frac(cancel(2) · 2 ( 1 − sqrt(3) ), cancel(2))$]],
  [Simplify.], [#math.equation(block: false, alt: "2 open parenthesis 1 minus the square root of 3 close parenthesis")[$2 ( 1 − sqrt(3) )$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 10 minus the square root of 75 over 5")[$frac(10 − sqrt(75), 5)$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 6 minus the square root of 45 over 3")[$frac(6 − sqrt(45), 3)$].

#solutionbox[
#math.equation(block: true, alt: "2 minus the square root of 5")[$2 − sqrt(5)$]
]
]


=== Use the Quotient Property to Simplify Square Roots

Whenever you have to simplify a square root, the first step you should take is to determine whether the radicand is a perfect square. A #emph[perfect square fraction] is a fraction in which both the numerator and the denominator are perfect squares.

#examplebox("Example 9")[][
Simplify: #math.equation(block: false, alt: "the square root of the fraction 9 over 64")[$sqrt(frac(9, 64))$].

#solutionbox[
#math.equation(block: true, alt: "the square root of the fraction 9 over 64; Since open parenthesis the fraction 3 over 8 close parenthesis squared equals the fraction 9 over 64, the fraction 3 over 8")[$&  &  & #h(4em) sqrt(frac(9, 64)) \ "Since" #h(0.2em) attach(( frac(3, 8) ), t: 2) = frac(9, 64) &  &  & #h(4em) frac(3, 8)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 25 over 16")[$sqrt(frac(25, 16))$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 49 over 81")[$sqrt(frac(49, 81))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 7 over 9")[$frac(7, 9)$]
]
]

If the numerator and denominator have any common factors, remove them. You may find a perfect square fraction!

#examplebox("Example 10")[][
Simplify: #math.equation(block: false, alt: "the square root of the fraction 45 over 80")[$sqrt(frac(45, 80))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of the fraction 45 over 80")[$sqrt(frac(45, 80))$]]),
  [Simplify inside the radical first. Rewrite #linebreak() showing the common factors of the #linebreak() numerator and denominator.], [#math.equation(block: false, alt: "the square root of the fraction 5 times 9 over 5 times 16")[$sqrt(frac(5 · 9, 5 · 16))$]],
  [Simplify the fraction by removing common factors.], [#math.equation(block: false, alt: "the square root of the fraction 9 over 16")[$sqrt(frac(9, 16))$]],
  [#math.equation(block: false, alt: "Simplify. open parenthesis the fraction 3 over 4 close parenthesis squared equals the fraction 9 over 16")[$"Simplify." #h(0.2em) attach(( frac(3, 4) ), t: 2) = frac(9, 16)$]], [#math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 75 over 48")[$sqrt(frac(75, 48))$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 98 over 162")[$sqrt(frac(98, 162))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 7 over 9")[$frac(7, 9)$]
]
]

In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the next example we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents, #math.equation(block: false, alt: "the fraction a to the power m over a to the power n equals a to the power m minus n , a not equal to 0")[$frac(a^(m), a^(n)) = a^(m − n) , a ≠ 0$].

#examplebox("Example 11")[][
Simplify: #math.equation(block: false, alt: "the square root of the fraction m to the power 6 over m to the power 4")[$sqrt(frac(m^(6), m^(4)))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of the fraction m to the power 6 over m to the power 4")[$sqrt(frac(m^(6), m^(4)))$]]),
  [Simplify the fraction inside the radical first.], [],
  [Divide the like bases by subtracting the exponents.], [#math.equation(block: false, alt: "the square root of m squared")[$sqrt(m^(2))$]],
  [Simplify.], [#math.equation(block: false, alt: "m")[$m$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction a to the power 8 over a to the power 6")[$sqrt(frac(a^(8), a^(6)))$].

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

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

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

#examplebox("Example 12")[][
Simplify: #math.equation(block: false, alt: "the square root of the fraction 48 p to the power 7 over 3 p cubed")[$sqrt(frac(48 p^(7), 3 p^(3)))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of the fraction 48 p to the power 7 over 3 p cubed")[$sqrt(frac(48 p^(7), 3 p^(3)))$]]),
  [Simplify the fraction inside the radical first.], [#math.equation(block: false, alt: "the square root of 16 p to the power 4")[$sqrt(16 p^(4))$]],
  [Simplify.], [#math.equation(block: false, alt: "4 p squared")[$4 p^(2)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 75 x to the power 5 over 3 x")[$sqrt(frac(75 x^(5), 3 x))$].

#solutionbox[
#math.equation(block: true, alt: "5 x squared")[$5 x^(2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 72 z to the power 12 over 2 z to the power 10")[$sqrt(frac(72 z^(12), 2 z^(10)))$].

#solutionbox[
#math.equation(block: true, alt: "6 z")[$6 z$]
]
]

Remember the Quotient to a Power Property? It said we could raise a fraction to a power by raising the numerator and denominator to the power separately.

#math.equation(block: true, alt: "open parenthesis the fraction a over b close parenthesis to the power m equals the fraction a to the power m over b to the power m , b not equal to 0")[$attach(( frac(a, b) ), t: m) = frac(a^(m), b^(m)) , b ≠ 0$]We can use a similar property to simplify a square root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect square we simplify the numerator and denominator separately.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Quotient Property of Square Roots]

If #emph[a], #emph[b] are non-negative real numbers and #math.equation(block: false, alt: "b not equal to 0")[$b ≠ 0$], then

#math.equation(block: true, alt: "the square root of the fraction a over b equals the fraction the square root of a over the square root of b")[$sqrt(frac(a, b)) = frac(sqrt(a), sqrt(b))$]
]

#examplebox("Example 13")[][
Simplify: #math.equation(block: false, alt: "the square root of the fraction 21 over 64")[$sqrt(frac(21, 64))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of the fraction 21 over 64")[$sqrt(frac(21, 64))$]]),
  [We cannot simplify the fraction inside the #linebreak() radical. Rewrite using the quotient property.], [#math.equation(block: false, alt: "the fraction the square root of 21 over the square root of 64")[$frac(sqrt(21), sqrt(64))$]],
  [Simplify the square root of 64. The #linebreak() numerator cannot be simplified.], [#math.equation(block: false, alt: "the fraction the square root of 21 over 8")[$frac(sqrt(21), 8)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 19 over 49")[$sqrt(frac(19, 49))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 19 over 7")[$frac(sqrt(19), 7)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 28 over 81")[$sqrt(frac(28, 81))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 the square root of 7 over 9")[$frac(2 sqrt(7), 9)$]
]
]

#examplebox("Example 14")[How to Use the Quotient Property to Simplify a Square Root][
Simplify: #math.equation(block: false, alt: "the square root of the fraction 27 m cubed over 196")[$sqrt(frac(27 m^(3), 196))$].

#solutionbox[
#figure(figph[This table has three columns and three rows. The first row reads, “Step 1. Simplify the fraction in the radicand, if possible.” Then it shows that 27 m cubed over 196 cannot be simplified. Then it shows the square root of 27 m cubed over 196.], alt: "This table has three columns and three rows. The first row reads, “Step 1. Simplify the fraction in the radicand, if possible.” Then it shows that 27 m cubed over 196 cannot be simplified. Then it shows the square root of 27 m cubed over 196.", caption: none)

#figure(figph[The second row says, “Step 2. Use the Quotient Property to rewrite the radical as the quotient of two radicals.” Then it says, “We rewrite the square root of 27 m cubed over 196 as the quotient of the square root of 27 m cubed and the square root of 196.” Then it shows the square root of 27 m cubed over the square root of 196.], alt: "The second row says, “Step 2. Use the Quotient Property to rewrite the radical as the quotient of two radicals.” Then it says, “We rewrite the square root of 27 m cubed over 196 as the quotient of the square root of 27 m cubed and the square root of 196.” Then it shows the square root of 27 m cubed over the square root of 196.", caption: none)

#figure(figph[The third row says, “Step 3. Simplify the radicals in the numerator and the denominator.” Then it says, “9 m squared and 196 are perfect squares.” It then shows the square root of 9 m squared time the square root of 3 m over the square root of 196. It then shows 3 m times the square root of 3 m over 14.], alt: "The third row says, “Step 3. Simplify the radicals in the numerator and the denominator.” Then it says, “9 m squared and 196 are perfect squares.” It then shows the square root of 9 m squared time the square root of 3 m over the square root of 196. It then shows 3 m times the square root of 3 m over 14.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 24 p cubed over 49")[$sqrt(frac(24 p^(3), 49))$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 48 x to the power 5 over 100")[$sqrt(frac(48 x^(5), 100))$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Simplify a square root using the quotient property.]

+ Simplify the fraction in the radicand, if possible.
+ Use the Quotient Property to rewrite the radical as the quotient of two radicals.
+ Simplify the radicals in the numerator and the denominator.
]

#examplebox("Example 15")[][
Simplify: #math.equation(block: false, alt: "the square root of the fraction 45 x to the power 5 over y to the power 4")[$sqrt(frac(45 x^(5), y^(4)))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of the fraction 45 x to the power 5 over y to the power 4")[$sqrt(frac(45 x^(5), y^(4)))$]]),
  [We cannot simplify the fraction in the #linebreak() radicand. Rewrite using the Quotient Property.], [#math.equation(block: false, alt: "the fraction the square root of 45 x to the power 5 over the square root of y to the power 4")[$frac(sqrt(45 x^(5)), sqrt(y^(4)))$]],
  [Simplify the radicals in the numerator and the denominator.], [#math.equation(block: false, alt: "the fraction the square root of 9 x to the power 4 times the square root of 5 x over y squared")[$frac(sqrt(9 x^(4)) · sqrt(5 x), y^(2))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 3 x squared the square root of 5 x over y squared")[$frac(3 x^(2) sqrt(5 x), y^(2))$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 80 m cubed over n to the power 6")[$sqrt(frac(80 m^(3), n^(6)))$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 54 u to the power 7 over v to the power 8")[$sqrt(frac(54 u^(7), v^(8)))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 u cubed the square root of 6 u over v to the power 4")[$frac(3 u^(3) sqrt(6 u), v^(4))$]
]
]

Be sure to simplify the fraction in the radicand first, if possible.

#examplebox("Example 16")[][
Simplify: #math.equation(block: false, alt: "the square root of the fraction 81 d to the power 9 over 25 d to the power 4")[$sqrt(frac(81 d^(9), 25 d^(4)))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of the fraction 81 d to the power 9 over 25 d to the power 4")[$sqrt(frac(81 d^(9), 25 d^(4)))$]]),
  [Simplify the fraction in the radicand.], [#math.equation(block: false, alt: "the square root of the fraction 81 d to the power 5 over 25")[$sqrt(frac(81 d^(5), 25))$]],
  [Rewrite using the Quotient Property.], [#math.equation(block: false, alt: "the fraction the square root of 81 d to the power 5 over the square root of 25")[$frac(sqrt(81 d^(5)), sqrt(25))$]],
  [Simplify the radicals in the numerator and the denominator.], [#math.equation(block: false, alt: "the fraction the square root of 81 d to the power 4 times the square root of d over 5")[$frac(sqrt(81 d^(4)) · sqrt(d), 5)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 9 d squared the square root of d over 5")[$frac(9 d^(2) sqrt(d), 5)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 64 x to the power 7 over 9 x cubed")[$sqrt(frac(64 x^(7), 9 x^(3)))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 8 x squared over 3")[$frac(8 x^(2), 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 16 a to the power 9 over 100 a to the power 5")[$sqrt(frac(16 a^(9), 100 a^(5)))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 a squared over 5")[$frac(2 a^(2), 5)$]
]
]

#examplebox("Example 17")[][
Simplify: #math.equation(block: false, alt: "the square root of the fraction 18 p to the power 5 q to the power 7 over 32 p q squared")[$sqrt(frac(18 p^(5) q^(7), 32 p q^(2)))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of the fraction 18 p to the power 5 q to the power 7 over 32 p q squared")[$sqrt(frac(18 p^(5) q^(7), 32 p q^(2)))$]]),
  [Simplify the fraction in the radicand, if possible.], [#math.equation(block: false, alt: "the square root of the fraction 9 p to the power 4 q to the power 5 over 16")[$sqrt(frac(9 p^(4) q^(5), 16))$]],
  [Rewrite using the Quotient Property.], [#math.equation(block: false, alt: "the fraction the square root of 9 p to the power 4 q to the power 5 over the square root of 16")[$frac(sqrt(9 p^(4) q^(5)), sqrt(16))$]],
  [Simplify the radicals in the numerator and the denominator.], [#math.equation(block: false, alt: "the fraction the square root of 9 p to the power 4 q to the power 4 times the square root of q over 4")[$frac(sqrt(9 p^(4) q^(4)) · sqrt(q), 4)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 3 p squared q squared the square root of q over 4")[$frac(3 p^(2) q^(2) sqrt(q), 4)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 50 x to the power 5 y cubed over 72 x to the power 4 y")[$sqrt(frac(50 x^(5) y^(3), 72 x^(4) y))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 y the square root of x over 6")[$frac(5 y sqrt(x), 6)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 48 m to the power 7 n squared over 125 m to the power 5 n to the power 9")[$sqrt(frac(48 m^(7) n^(2), 125 m^(5) n^(9)))$].

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


=== Key Concepts

- #strong[Simplified Square Root] #math.equation(block: false, alt: "the square root of a")[$sqrt(a)$] is considered simplified if #math.equation(block: false, alt: "a")[$a$] has no perfect-square factors.
- #strong[Product Property of Square Roots] If #emph[a], #emph[b] are non-negative real numbers, then #linebreak() 
#math.equation(block: true, alt: "the square root of a b equals the square root of a times the square root of b")[$sqrt(a b) = sqrt(a) · sqrt(b)$]
- #strong[Simplify a Square Root Using the Product Property] To simplify a square root using the Product Property:
+ Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect square factor.
+ Use the product rule to rewrite the radical as the product of two radicals.
+ Simplify the square root of the perfect square.
- #strong[Quotient Property of Square Roots] If #emph[a], #emph[b] are non-negative real numbers and #math.equation(block: false, alt: "b not equal to 0")[$b ≠ 0$], then #linebreak() 
#math.equation(block: true, alt: "the square root of the fraction a over b equals the fraction the square root of a over the square root of b")[$sqrt(frac(a, b)) = frac(sqrt(a), sqrt(b))$] #linebreak()  #linebreak()  #linebreak()
- #strong[Simplify a Square Root Using the Quotient Property] To simplify a square root using the Quotient Property:
+ Simplify the fraction in the radicand, if possible.
+ Use the Quotient Rule to rewrite the radical as the quotient of two radicals.
+ Simplify the radicals in the numerator and the denominator.


==== Practice Makes Perfect

#strong[Use the Product Property to Simplify Square Roots]

In the following exercises, simplify.

#math.equation(block: true, alt: "the square root of 27")[$sqrt(27)$]

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

#math.equation(block: true, alt: "the square root of 80")[$sqrt(80)$]

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

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

#math.equation(block: true, alt: "the square root of 96")[$sqrt(96)$]

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

#solutionbox[
#math.equation(block: true, alt: "10 the square root of 2")[$10 sqrt(2)$]
]

#math.equation(block: true, alt: "the square root of 147")[$sqrt(147)$]

#math.equation(block: true, alt: "the square root of 450")[$sqrt(450)$]

#solutionbox[
#math.equation(block: true, alt: "15 the square root of 2")[$15 sqrt(2)$]
]

#math.equation(block: true, alt: "the square root of 252")[$sqrt(252)$]

#math.equation(block: true, alt: "the square root of 800")[$sqrt(800)$]

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

#math.equation(block: true, alt: "the square root of 288")[$sqrt(288)$]

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

#solutionbox[
#math.equation(block: true, alt: "15 the square root of 3")[$15 sqrt(3)$]
]

#math.equation(block: true, alt: "the square root of 1250")[$sqrt(1250)$]

#math.equation(block: true, alt: "the square root of x to the power 7")[$sqrt(x^(7))$]

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

#math.equation(block: true, alt: "the square root of y to the power 11")[$sqrt(y^(11))$]

#math.equation(block: true, alt: "the square root of p cubed")[$sqrt(p^(3))$]

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

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

#math.equation(block: true, alt: "the square root of m to the power 13")[$sqrt(m^(13))$]

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

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

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

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

#math.equation(block: true, alt: "the square root of s to the power 33")[$sqrt(s^(33))$]

#math.equation(block: true, alt: "the square root of 49 n to the power 17")[$sqrt(49 n^(17))$]

#solutionbox[
#math.equation(block: true, alt: "7 n to the power 8 the square root of n")[$7 n^(8) sqrt(n)$]
]

#math.equation(block: true, alt: "the square root of 25 m to the power 9")[$sqrt(25 m^(9))$]

#math.equation(block: true, alt: "the square root of 81 r to the power 15")[$sqrt(81 r^(15))$]

#solutionbox[
#math.equation(block: true, alt: "9 r to the power 7 the square root of r")[$9 r^(7) sqrt(r)$]
]

#math.equation(block: true, alt: "the square root of 100 s to the power 19")[$sqrt(100 s^(19))$]

#math.equation(block: true, alt: "the square root of 98 m to the power 5")[$sqrt(98 m^(5))$]

#solutionbox[
#math.equation(block: true, alt: "7 m squared the square root of 2 m")[$7 m^(2) sqrt(2 m)$]
]

#math.equation(block: true, alt: "the square root of 32 n to the power 11")[$sqrt(32 n^(11))$]

#math.equation(block: true, alt: "the square root of 125 r to the power 13")[$sqrt(125 r^(13))$]

#solutionbox[
#math.equation(block: true, alt: "5 r to the power 6 the square root of 5 r")[$5 r^(6) sqrt(5 r)$]
]

#math.equation(block: true, alt: "the square root of 80 s to the power 15")[$sqrt(80 s^(15))$]

#math.equation(block: true, alt: "the square root of 200 p to the power 13")[$sqrt(200 p^(13))$]

#solutionbox[
#math.equation(block: true, alt: "10 p to the power 6 the square root of 2 p")[$10 p^(6) sqrt(2 p)$]
]

#math.equation(block: true, alt: "the square root of 128 q cubed")[$sqrt(128 q^(3))$]

#math.equation(block: true, alt: "the square root of 242 m to the power 23")[$sqrt(242 m^(23))$]

#solutionbox[
#math.equation(block: true, alt: "11 m to the power 11 the square root of 2 m")[$11 m^(11) sqrt(2 m)$]
]

#math.equation(block: true, alt: "the square root of 175 n to the power 13")[$sqrt(175 n^(13))$]

#math.equation(block: true, alt: "the square root of 147 m to the power 7 n to the power 11")[$sqrt(147 m^(7) n^(11))$]

#solutionbox[
#math.equation(block: true, alt: "7 m cubed n to the power 5 the square root of 3 m n")[$7 m^(3) n^(5) sqrt(3 m n)$]
]

#math.equation(block: true, alt: "the square root of 48 m to the power 7 n to the power 5")[$sqrt(48 m^(7) n^(5))$]

#math.equation(block: true, alt: "the square root of 75 r to the power 13 s to the power 9")[$sqrt(75 r^(13) s^(9))$]

#solutionbox[
#math.equation(block: true, alt: "5 r to the power 6 s to the power 4 the square root of 3 r s")[$5 r^(6) s^(4) sqrt(3 r s)$]
]

#math.equation(block: true, alt: "the square root of 96 r cubed s cubed")[$sqrt(96 r^(3) s^(3))$]

#math.equation(block: true, alt: "the square root of 300 p to the power 9 q to the power 11")[$sqrt(300 p^(9) q^(11))$]

#solutionbox[
#math.equation(block: true, alt: "10 p to the power 4 q to the power 5 the square root of 3 p q")[$10 p^(4) q^(5) sqrt(3 p q)$]
]

#math.equation(block: true, alt: "the square root of 192 q cubed r to the power 7")[$sqrt(192 q^(3) r^(7))$]

#math.equation(block: true, alt: "the square root of 242 m to the power 13 n to the power 21")[$sqrt(242 m^(13) n^(21))$]

#solutionbox[
#math.equation(block: true, alt: "11 m to the power 6 n to the power 10 the square root of 2 m n")[$11 m^(6) n^(10) sqrt(2 m n)$]
]

#math.equation(block: true, alt: "the square root of 150 m to the power 9 n cubed")[$sqrt(150 m^(9) n^(3))$]

#math.equation(block: true, alt: "5 plus the square root of 12")[$5 + sqrt(12)$]

#solutionbox[
#math.equation(block: true, alt: "5 plus 2 the square root of 3")[$5 + 2 sqrt(3)$]
]

#math.equation(block: true, alt: "8 plus the square root of 96")[$8 + sqrt(96)$]

#math.equation(block: true, alt: "1 plus the square root of 45")[$1 + sqrt(45)$]

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

#math.equation(block: true, alt: "3 plus the square root of 125")[$3 + sqrt(125)$]

#math.equation(block: true, alt: "the fraction 10 minus the square root of 24 over 2")[$frac(10 − sqrt(24), 2)$]

#solutionbox[
#math.equation(block: true, alt: "5 minus the square root of 6")[$5 − sqrt(6)$]
]

#math.equation(block: true, alt: "the fraction 8 minus the square root of 80 over 4")[$frac(8 − sqrt(80), 4)$]

#math.equation(block: true, alt: "the fraction 3 plus the square root of 90 over 3")[$frac(3 + sqrt(90), 3)$]

#solutionbox[
#math.equation(block: true, alt: "1 plus the square root of 10")[$1 + sqrt(10)$]
]

#math.equation(block: true, alt: "the fraction 15 plus the square root of 75 over 5")[$frac(15 + sqrt(75), 5)$]

#strong[Use the Quotient Property to Simplify Square Roots]

In the following exercises, simplify.

#math.equation(block: true, alt: "the square root of the fraction 49 over 64")[$sqrt(frac(49, 64))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 7 over 8")[$frac(7, 8)$]
]

#math.equation(block: true, alt: "the square root of the fraction 100 over 36")[$sqrt(frac(100, 36))$]

#math.equation(block: true, alt: "the square root of the fraction 121 over 16")[$sqrt(frac(121, 16))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 11 over 4")[$frac(11, 4)$]
]

#math.equation(block: true, alt: "the square root of the fraction 144 over 169")[$sqrt(frac(144, 169))$]

#math.equation(block: true, alt: "the square root of the fraction 72 over 98")[$sqrt(frac(72, 98))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 6 over 7")[$frac(6, 7)$]
]

#math.equation(block: true, alt: "the square root of the fraction 75 over 12")[$sqrt(frac(75, 12))$]

#math.equation(block: true, alt: "the square root of the fraction 45 over 125")[$sqrt(frac(45, 125))$]

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

#math.equation(block: true, alt: "the square root of the fraction 300 over 243")[$sqrt(frac(300, 243))$]

#math.equation(block: true, alt: "the square root of the fraction x to the power 10 over x to the power 6")[$sqrt(frac(x^(10), x^(6)))$]

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

#math.equation(block: true, alt: "the square root of the fraction p to the power 20 over p to the power 10")[$sqrt(frac(p^(20), p^(10)))$]

#math.equation(block: true, alt: "the square root of the fraction y to the power 4 over y to the power 8")[$sqrt(frac(y^(4), y^(8)))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over y squared")[$frac(1, y^(2))$]
]

#math.equation(block: true, alt: "the square root of the fraction q to the power 8 over q to the power 14")[$sqrt(frac(q^(8), q^(14)))$]

#math.equation(block: true, alt: "the square root of the fraction 200 x to the power 7 over 2 x cubed")[$sqrt(frac(200 x^(7), 2 x^(3)))$]

#solutionbox[
#math.equation(block: true, alt: "10 x squared")[$10 x^(2)$]
]

#math.equation(block: true, alt: "the square root of the fraction 98 y to the power 11 over 2 y to the power 5")[$sqrt(frac(98 y^(11), 2 y^(5)))$]

#math.equation(block: true, alt: "the square root of the fraction 96 p to the power 9 over 6 p")[$sqrt(frac(96 p^(9), 6 p))$]

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

#math.equation(block: true, alt: "the square root of the fraction 108 q to the power 10 over 3 q squared")[$sqrt(frac(108 q^(10), 3 q^(2)))$]

#math.equation(block: true, alt: "the square root of the fraction 36 over 35")[$sqrt(frac(36, 35))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 6 over the square root of 35")[$frac(6, sqrt(35))$]
]

#math.equation(block: true, alt: "the square root of the fraction 144 over 65")[$sqrt(frac(144, 65))$]

#math.equation(block: true, alt: "the square root of the fraction 20 over 81")[$sqrt(frac(20, 81))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 the square root of 5 over 9")[$frac(2 sqrt(5), 9)$]
]

#math.equation(block: true, alt: "the square root of the fraction 21 over 196")[$sqrt(frac(21, 196))$]

#math.equation(block: true, alt: "the square root of the fraction 96 x to the power 7 over 121")[$sqrt(frac(96 x^(7), 121))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 x cubed the square root of 6 x over 11")[$frac(4 x^(3) sqrt(6 x), 11)$]
]

#math.equation(block: true, alt: "the square root of the fraction 108 y to the power 4 over 49")[$sqrt(frac(108 y^(4), 49))$]

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

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

#math.equation(block: true, alt: "the square root of the fraction 125 n to the power 7 over 169")[$sqrt(frac(125 n^(7), 169))$]

#math.equation(block: true, alt: "the square root of the fraction 98 r to the power 5 over 100")[$sqrt(frac(98 r^(5), 100))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 7 r squared the square root of 2 r over 10")[$frac(7 r^(2) sqrt(2 r), 10)$]
]

#math.equation(block: true, alt: "the square root of the fraction 180 s to the power 10 over 144")[$sqrt(frac(180 s^(10), 144))$]

#math.equation(block: true, alt: "the square root of the fraction 28 q to the power 6 over 225")[$sqrt(frac(28 q^(6), 225))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 q cubed the square root of 7 over 15")[$frac(2 q^(3) sqrt(7), 15)$]
]

#math.equation(block: true, alt: "the square root of the fraction 150 r cubed over 256")[$sqrt(frac(150 r^(3), 256))$]

#math.equation(block: true, alt: "the square root of the fraction 75 r to the power 9 over s to the power 8")[$sqrt(frac(75 r^(9), s^(8)))$]

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

#math.equation(block: true, alt: "the square root of the fraction 72 x to the power 5 over y to the power 6")[$sqrt(frac(72 x^(5), y^(6)))$]

#math.equation(block: true, alt: "the square root of the fraction 28 p to the power 7 over q squared")[$sqrt(frac(28 p^(7), q^(2)))$]

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

#math.equation(block: true, alt: "the square root of the fraction 45 r cubed over s to the power 10")[$sqrt(frac(45 r^(3), s^(10)))$]

#math.equation(block: true, alt: "the square root of the fraction 100 x to the power 5 over 36 x cubed")[$sqrt(frac(100 x^(5), 36 x^(3)))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 x over 3")[$frac(5 x, 3)$]
]

#math.equation(block: true, alt: "the square root of the fraction 49 r to the power 12 over 16 r to the power 6")[$sqrt(frac(49 r^(12), 16 r^(6)))$]

#math.equation(block: true, alt: "the square root of the fraction 121 p to the power 5 over 81 p squared")[$sqrt(frac(121 p^(5), 81 p^(2)))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 11 p the square root of p over 9")[$frac(11 p sqrt(p), 9)$]
]

#math.equation(block: true, alt: "the square root of the fraction 25 r to the power 8 over 64 r")[$sqrt(frac(25 r^(8), 64 r))$]

#math.equation(block: true, alt: "the square root of the fraction 32 x to the power 5 y cubed over 18 x cubed y")[$sqrt(frac(32 x^(5) y^(3), 18 x^(3) y))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 x y over 3")[$frac(4 x y, 3)$]
]

#math.equation(block: true, alt: "the square root of the fraction 75 r to the power 6 s to the power 8 over 48 r s to the power 4")[$sqrt(frac(75 r^(6) s^(8), 48 r s^(4)))$]

#math.equation(block: true, alt: "the square root of the fraction 27 p squared q over 108 p to the power 5 q cubed")[$sqrt(frac(27 p^(2) q, 108 p^(5) q^(3)))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 2 p q the square root of p")[$frac(1, 2 p q sqrt(p))$]
]

#math.equation(block: true, alt: "the square root of the fraction 50 r to the power 5 s squared over 128 r squared s to the power 5")[$sqrt(frac(50 r^(5) s^(2), 128 r^(2) s^(5)))$]


==== Everyday Math

+ ⓐ Elliott decides to construct a square garden that will take up 288 square feet of his yard. Simplify #math.equation(block: false, alt: "the square root of 288")[$sqrt(288)$] to determine the length and the width of his garden. Round to the nearest tenth of a foot.
+ ⓑ Suppose Elliott decides to reduce the size of his square garden so that he can create a 5-foot-wide walking path on the north and east sides of the garden. Simplify #math.equation(block: false, alt: "the square root of 288 minus 5")[$sqrt(288) − 5$] to determine the length and width of the new garden. Round to the nearest tenth of a foot.

#solutionbox[
ⓐ #math.equation(block: false, alt: "17.0 feet")[$17.0 #h(0.2em) "feet"$] ⓑ #math.equation(block: false, alt: "12.0 feet")[$12.0 #h(0.2em) "feet"$]
]

+ ⓐ Melissa accidentally drops a pair of sunglasses from the top of a roller coaster, 64 feet above the ground. Simplify #math.equation(block: false, alt: "the square root of the fraction 64 over 16")[$sqrt(frac(64, 16))$] to determine the number of seconds it takes for the sunglasses to reach the ground.
+ ⓑ Suppose the sunglasses in the previous example were dropped from a height of 144 feet. Simplify #math.equation(block: false, alt: "the square root of the fraction 144 over 16")[$sqrt(frac(144, 16))$] to determine the number of seconds it takes for the sunglasses to reach the ground.


==== Writing Exercises

Explain why #math.equation(block: false, alt: "the square root of x to the power 4 equals x squared")[$sqrt(x^(4)) = x^(2)$]. Then explain why #math.equation(block: false, alt: "the square root of x to the power 16 equals x to the power 8")[$sqrt(x^(16)) = x^(8)$].

#solutionbox[
Answers will vary.
]

Explain why #math.equation(block: false, alt: "7 plus the square root of 9")[$7 + sqrt(9)$] is not equal to #math.equation(block: false, alt: "the square root of 7 plus 9")[$sqrt(7 + 9)$].


==== 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 three rows. The columns are labeled, “I can…,” “confidently,” “with some help,” and “no—I don’t get it!” The rows under “I can…” Read, “use the Product Property to simplify square roots.,” and “use the Quotient Property to simplify square roots.” The other rows unders the other columns are blank.], alt: "This table has four columns and three rows. The columns are labeled, “I can…,” “confidently,” “with some help,” and “no—I don’t get it!” The rows under “I can…” Read, “use the Product Property to simplify square roots.,” and “use the Quotient Property to simplify square roots.” The other rows unders the other columns are blank.", caption: none)

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