7.5 Simplify Square Roots

Izabela Mazur

CHAPTER 7 Powers, Roots, and Scientific Notation

7.5 Simplify Square Roots

Learning Objectives

By the end of this section, you will be able to:

Use the Product Property to simplify square roots
Use the Quotient Property to simplify square roots

In the last section, we estimated the square root of a number between two consecutive whole numbers. We can say that $\sqrt{50}$ is between 7 and 8. This is fairly easy to do when the numbers are small enough that we can use in (Simplify and Use Square Roots).

But what if we want to estimate $\sqrt{500}$ ? 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 simplified if its radicand contains no perfect square factors.

Simplified Square Root

$\sqrt{a}$ is considered simplified if $a$ has no perfect square factors.

So $\sqrt{31}$ is simplified. But $\sqrt{32}$ is not simplified, because 16 is a perfect square factor of 32

The properties we will use to simplify expressions with square roots are similar to the properties of exponents. We know that ${\left(ab\right)}^{m}={a}^{m}{b}^{m}$ . The corresponding property of square roots says that $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ .

Product Property of Square Roots

If a, b are non-negative real numbers, then $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ .

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 (Example 1).

EXAMPLE 1

How To Use the Product Property to Simplify a Square Root

Simplify: $\sqrt{50}$ .

Solution

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. 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. 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.

TRY IT 1.1

Simplify: $\sqrt{48}$ .

Show answer

$4\sqrt{3}$

TRY IT 1.2

Simplify: $\sqrt{45}$ .

Show answer

$3\sqrt{5}$

Notice in the previous example that the simplified form of $\sqrt{50}$ is $5\sqrt{2}$ , which is the product of an integer and a square root. We always write the integer in front of the square root.

HOW 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.

EXAMPLE 2

Simplify: $\sqrt{500}$ .

Solution

	$\sqrt{500}$
Rewrite the radicand as a product using the largest perfect square factor.	$\sqrt{100\cdot 5}$
Rewrite the radical as the product of two radicals.	$\sqrt{100}\cdot \sqrt{5}$
Simplify.	$10\sqrt{5}$

TRY IT 2.1

Simplify: $\sqrt{288}$ .

Show answer

$12\sqrt{2}$

TRY IT 2.2

Simplify: $\sqrt{432}$ .

Show answer

$12\sqrt{3}$

We could use the simplified form $10\sqrt{5}$ to estimate $\sqrt{500}$ . We know 5 is between 2 and 3, and $\sqrt{500}$ is $10\sqrt{5}$ . So $\sqrt{500}$ is between 20 and 30.

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

EXAMPLE 3

Simplify: $\sqrt{{x}^{3}}$ .

Solution

	$\sqrt{{x}^{3}}$
Rewrite the radicand as a product using the largest perfect square factor.	$\sqrt{{x}^{2}\cdot x}$
Rewrite the radical as the product of two radicals.	$\sqrt{{x}^{2}}\cdot\sqrt{x}$
Simplify.	$x\sqrt{x}$

TRY IT 3.1

Simplify: $\sqrt{{b}^{5}}$ .

Show answer

${b}^{2}\sqrt{b}$

TRY IT 3.2

Simplify: $\sqrt{{p}^{9}}$ .

Show answer

${p}^{4}\sqrt{p}$

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

EXAMPLE 4

Simplify: $\sqrt{25{y}^{5}.}$

Solution

	$\sqrt{25{y}^{5}}$
Rewrite the radicand as a product using the largest perfect square factor.	$\sqrt{25{y}^{4}\cdot y}$
Rewrite the radical as the product of two radicals.	$\sqrt{25{y}^{4}}\cdot\sqrt{y}$
Simplify.	$5{y}^{2}\sqrt{y}$

TRY IT 4.1

Simplify: $\sqrt{16{x}^{7}}$ .

Show answer

$4{x}^{3}\sqrt{x}$

TRY IT 4.2

Simplify: $\sqrt{49{v}^{9}}$ .

Show answer

$7{v}^{4}\sqrt{v}$

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

EXAMPLE 5

Simplify: $\sqrt{72{n}^{7}}$ .

Solution

	$\sqrt{72{n}^{7}}$
Rewrite the radicand as a product using the largest perfect square factor.	$\sqrt{36{n}^{6}\cdot 2n}$
Rewrite the radical as the product of two radicals.	$\sqrt{36{n}^{6}}\cdot \sqrt{2n}$
Simplify.	$6{n}^{3}\sqrt{2n}$

TRY IT 5.1

Simplify: $\sqrt{32{y}^{5}}$ .

Show answer

$4{y}^{2}\sqrt{2y}$

TRY IT 5.2

Simplify: $\sqrt{75{a}^{9}}$ .

Show answer

$5{a}^{4}\sqrt{3a}$

EXAMPLE 6

Simplify: $\sqrt{63{u}^{3}{v}^{5}}$ .

Solution

	$\sqrt{63{u}^{3}{v}^{5}}$
Rewrite the radicand as a product using the largest perfect square factor.	$\sqrt{9{u}^{2}{v}^{4}\cdot 7uv}$
Rewrite the radical as the product of two radicals.	$\sqrt{9{u}^{2}{v}^{4}}\cdot\sqrt{7uv}$
Simplify.	$3u{v}^{2}\sqrt{7uv}$

TRY IT 6.1

Simplify: $\sqrt{98{a}^{7}{b}^{5}}$ .

Show answer

$7{a}^{3}{b}^{2}{\sqrt{2ab}}^{}$

TRY IT 6.2

Simplify: $\sqrt{180{m}^{9}{n}^{11}}$ .

Show answer

$6{m}^{4}{n}^{5}\sqrt{5mn}$

We have seen how to use the Order of Operations to simplify some expressions with radicals. To simplify $\sqrt{25}+\sqrt{144}$ we must simplify each square root separately first, then add to get the sum of 17

The expression $\sqrt{17}+\sqrt{7}$ 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.

EXAMPLE 7

Simplify: $3+\sqrt{32}$ .

Solution

	$3+\sqrt{32}$
Rewrite the radicand as a product using the largest perfect square factor.	$3+\sqrt{16\cdot 2}$
Rewrite the radical as the product of two radicals.	$3+\sqrt{16}\cdot\sqrt{2}$
Simplify.	$3+4\sqrt{2}$

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!

TRY IT 7.1

Simplify: $5+\sqrt{75}$ .

Show answer

$5+5\sqrt{3}$

TRY IT 7.2

Simplify: $2+\sqrt{98}$ .

Show answer

$2+7\sqrt{2}$

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.

EXAMPLE 8

Simplify: $\frac{4-\sqrt{48}}{2}$ .

Solution

	$\frac{4-\sqrt{48}}{2}$
Rewrite the radicand as a product using the largest perfect square factor.	$\frac{4-\sqrt{16\cdot 3}}{2}$
Rewrite the radical as the product of two radicals.	$\frac{4-\sqrt{16}\cdot\sqrt{3}}{2}$
Simplify.	$\frac{4-4\sqrt{3}}{2}$
Factor the common factor from the numerator.	$\frac{4\left(1-\sqrt{3}\right)}{2}$
Remove the common factor, 2, from the numerator and denominator.	$\frac{\overline{)2}\cdot 2\left(1-\sqrt{3}\right)}{\overline{)2}}$
Simplify.	$2\left(1-\sqrt{3}\right)$

TRY IT 8.1

Simplify: $\frac{10-\sqrt{75}}{5}$ .

Show answer

$2-\sqrt{3}$

TRY IT 8.2

Simplify: $\frac{6-\sqrt{45}}{3}$ .

Show answer

$2-\sqrt{5}$

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 perfect square fraction is a fraction in which both the numerator and the denominator are perfect squares.

EXAMPLE 9

Simplify: $\sqrt{\frac{9}{64}}$ .

Solution

	$\sqrt{\frac{9}{64}}$
$\text{Since}{\left(\frac{3}{8}\right)}^{2}=\frac{9}{64}$	$\frac{3}{8}$

TRY IT 9.1

Simplify: $\sqrt{\frac{25}{16}}$ .

Show answer

$\frac{5}{4}$

TRY IT 9.2

Simplify: $\sqrt{\frac{49}{81}}$ .

Show answer

$\frac{7}{9}$

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

EXAMPLE 10

Simplify: $\sqrt{\frac{45}{80}}$ .

Solution

	$\sqrt{\frac{45}{80}}$
Simplify inside the radical first. Rewrite showing the common factors of the numerator and denominator.	$\sqrt{\frac{5\cdot 9}{5\cdot 16}}$
Simplify the fraction by removing common factors.	$\sqrt{\frac{9}{16}}$
$\text{Simplify}{\left(\frac{3}{4}\right)}^{2}=\frac{9}{16}$	$\frac{3}{4}$

TRY IT 10.1

Simplify: $\sqrt{\frac{75}{48}}$ .

Show answer

$\frac{5}{4}$

TRY IT 10.2

Simplify: $\sqrt{\frac{98}{162}}$ .

Show answer

$\frac{7}{9}$

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, $\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},a\ne 0$ .

EXAMPLE 11

Simplify: $\sqrt{\frac{{m}^{6}}{{m}^{4}}}$ .

Solution

	$\sqrt{\frac{{m}^{6}}{{m}^{4}}}$
Simplify the fraction inside the radical first. Divide the like bases by subtracting the exponents.	$\sqrt{{m}^{2}}$
Simplify.	$m$

TRY IT 11.1

Simplify: $\sqrt{\frac{{a}^{8}}{{a}^{6}}}$ .

Show answer

$a$

TRY IT 11.2

Simplify: $\sqrt{\frac{{x}^{14}}{{x}^{10}}}$ .

Show answer

${x}^{2}$

EXAMPLE 12

Simplify: $\sqrt{\frac{48{p}^{7}}{3{p}^{3}}}$ .

Solution

	$\sqrt{\frac{48{p}^{7}}{3{p}^{3}}}$
Simplify the fraction inside the radical first.	$\sqrt{16{p}^{4}}$
Simplify.	$4{p}^{2}$

TRY IT 12.1

Simplify: $\sqrt{\frac{75{x}^{5}}{3x}}$ .

Show answer

$5{x}^{2}$

TRY IT 12.2

Simplify: $\sqrt{\frac{72{z}^{12}}{2{z}^{10}}}$ .

Show answer

$6z$

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.

${\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}},b\ne 0$

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.

Quotient Property of Square Roots

If a, b are non-negative real numbers and $b\ne 0$ , then

$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$

EXAMPLE 13

Simplify: $\sqrt{\frac{21}{64}}$ .

Solution

	$\sqrt{\frac{21}{64}}$
We cannot simplify the fraction inside the radical. Rewrite using the quotient property.	$\frac{\sqrt{21}}{\sqrt{64}}$
Simplify the square root of 64. The numerator cannot be simplified.	$\frac{\sqrt{21}}{8}$

TRY IT 13.1

Simplify: $\sqrt{\frac{19}{49}}$ .

Show answer

$\frac{\sqrt{19}}{7}$

TRY IT 13.2

Simplify: $\sqrt{\frac{28}{81}}$ .

Show answer

$\frac{2\sqrt{7}}{9}$

EXAMPLE 14

How to Use the Quotient Property to Simplify a Square Root

Simplify: $\sqrt{\frac{27{m}^{3}}{196}}$ .

Solution

$This table has three columns and three rows. The first row reads, “Step 1. Simplify the fraction in the radicand, if possible.” Then it shows that 27 m cubed over 196 cannot be simplified. Then it shows the square root of 27 m cubed over 196.$ 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. 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.

TRY IT 14.1

Simplify: $\sqrt{\frac{24{p}^{3}}{49}}$ .

Show answer

$\frac{2p\sqrt{6p}}{7}$

TRY IT 14.2

Simplify: $\sqrt{\frac{48{x}^{5}}{100}}$ .

Show answer

$\frac{2{x}^{2}\sqrt{3x}}{5}$

HOW TO: 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.

EXAMPLE 15

Simplify: $\sqrt{\frac{45{x}^{5}}{{y}^{4}}}$ .

Solution

	$\sqrt{\frac{45{x}^{5}}{{y}^{4}}}$
We cannot simplify the fraction in the radicand. Rewrite using the Quotient Property.	$\frac{\sqrt{45{x}^{5}}}{\sqrt{{y}^{4}}}$
Simplify the radicals in the numerator and the denominator.	$\frac{\sqrt{9{x}^{4}}\cdot\sqrt{5x}}{{y}^{2}}$
Simplify.	$\frac{3{x}^{2}\sqrt{5x}}{{y}^{2}}$

TRY IT 15.1

Simplify: $\sqrt{\frac{80{m}^{3}}{{n}^{6}}}$ .

Show answer

$\frac{4m\sqrt{5m}}{{n}^{3}}$

TRY IT 15.2

Simplify: $\sqrt{\frac{54{u}^{7}}{{v}^{8}}}$ .

Show answer

$\frac{3{u}^{3}\sqrt{6u}}{{v}^{4}}$

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

EXAMPLE 16

Simplify: $\sqrt{\frac{81{d}^{9}}{25{d}^{4}}}$ .

Solution

	$\sqrt{\frac{81{d}^{9}}{25{d}^{4}}}$
Simplify the fraction in the radicand.	$\sqrt{\frac{81{d}^{5}}{25}}$
Rewrite using the Quotient Property.	$\frac{\sqrt{81{d}^{5}}}{\sqrt{25}}$
Simplify the radicals in the numerator and the denominator.	$\frac{\sqrt{81{d}^{4}}\cdot\sqrt{d}}{5}$
Simplify.	$\frac{9{d}^{2}\sqrt{d}}{5}$

TRY IT 16.1

Simplify: $\sqrt{\frac{64{x}^{7}}{9{x}^{3}}}$ .

Show answer

$\frac{8{x}^{2}}{3}$

TRY IT 16.2

Simplify: $\sqrt{\frac{16{a}^{9}}{100{a}^{5}}}$ .

Show answer

$\frac{2{a}^{2}}{5}$

EXAMPLE 17

Simplify: $\sqrt{\frac{18{p}^{5}{q}^{7}}{32p{q}^{2}}}$ .

Solution

	$\sqrt{\frac{18{p}^{5}{q}^{7}}{32p{q}^{2}}}$
Simplify the fraction in the radicand, if possible.	$\sqrt{\frac{9{p}^{4}{q}^{5}}{16}}$
Rewrite using the Quotient Property.	$\frac{\sqrt{9{p}^{4}{q}^{5}}}{\sqrt{16}}$
Simplify the radicals in the numerator and the denominator.	$\frac{\sqrt{9{p}^{4}{q}^{4}}\cdot\sqrt{q}}{4}$
Simplify.	$\frac{3{p}^{2}{q}^{2}\sqrt{q}}{4}$

TRY IT 17.1

Simplify: $\sqrt{\frac{50{x}^{5}{y}^{3}}{72{x}^{4}y}}$ .

Show answer

$\frac{5y\sqrt{x}}{6}$

TRY IT 17.2

Simplify: $\sqrt{\frac{48{m}^{7}{n}^{2}}{125{m}^{5}{n}^{9}}}$ .

Show answer

$\frac{4m\sqrt{3}}{5{n}^{3}\sqrt{5n}}$

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1. $\sqrt{27}$	2. $\sqrt{80}$
3. $\sqrt{125}$	4. $\sqrt{96}$
5. $\sqrt{200}$	6. $\sqrt{147}$
7. $\sqrt{450}$	8. $\sqrt{252}$
9. $\sqrt{800}$	10. $\sqrt{288}$
11. $\sqrt{675}$	12. $\sqrt{1250}$
13. $\sqrt{{x}^{7}}$	14. $\sqrt{{y}^{11}}$
15. $\sqrt{{p}^{3}}$	16. $\sqrt{{q}^{5}}$
17. $\sqrt{{m}^{13}}$	18. $\sqrt{{n}^{21}}$
19. $\sqrt{{r}^{25}}$	20. $\sqrt{{s}^{33}}$
21. $\sqrt{49{n}^{17}}$	22. $\sqrt{25{m}^{9}}$
23. $\sqrt{81{r}^{15}}$	24. $\sqrt{100{s}^{19}}$
25. $\sqrt{98{m}^{5}}$	26. $\sqrt{32{n}^{11}}$
27. $\sqrt{125{r}^{13}}$	28. $\sqrt{80{s}^{15}}$
29. $\sqrt{200{p}^{13}}$	30. $\sqrt{128{q}^{3}}$
31. $\sqrt{242{m}^{23}}$	32. $\sqrt{175{n}^{13}}$
33. $\sqrt{147{m}^{7}{n}^{11}}$	34. $\sqrt{48{m}^{7}{n}^{5}}$
35. $\sqrt{75{r}^{13}{s}^{9}}$	36. $\sqrt{96{r}^{3}{s}^{3}}$
37. $\sqrt{300{p}^{9}{q}^{11}}$	38. $\sqrt{192{q}^{3}{r}^{7}}$
39. $\sqrt{242{m}^{13}{n}^{21}}$	40. $\sqrt{150{m}^{9}{n}^{3}}$
41. $5+\sqrt{12}$	42. $8+\sqrt{96}$
43. $1+\sqrt{45}$	44. $3+\sqrt{125}$
45. $\frac{10-\sqrt{24}}{2}$	46. $\frac{8-\sqrt{80}}{4}$
47. $\frac{3+\sqrt{90}}{3}$	48. $\frac{15+\sqrt{75}}{5}$

49. $\sqrt{\frac{49}{64}}$	50. $\sqrt{\frac{100}{36}}$
51. $\sqrt{\frac{121}{16}}$	52. $\sqrt{\frac{144}{169}}$
53. $\sqrt{\frac{72}{98}}$	54. $\sqrt{\frac{75}{12}}$
55. $\sqrt{\frac{9}{25}}$	56. $\sqrt{\frac{300}{243}}$
57. $\sqrt{\frac{{x}^{10}}{{x}^{6}}}$	58. $\sqrt{\frac{{p}^{20}}{{p}^{10}}}$
59. $\sqrt{\frac{{y}^{4}}{{y}^{8}}}$	60. $\sqrt{\frac{{q}^{8}}{{q}^{14}}}$
61. $\sqrt{\frac{200{x}^{7}}{2{x}^{3}}}$	62. $\sqrt{\frac{98{y}^{11}}{2{y}^{5}}}$
63. $\sqrt{\frac{96{p}^{9}}{6p}}$	64. $\sqrt{\frac{108{q}^{10}}{3{q}^{2}}}$
65. $\sqrt{\frac{36}{35}}$	66. $\sqrt{\frac{144}{65}}$
67. $\sqrt{\frac{20}{81}}$	68. $\sqrt{\frac{21}{196}}$
69. $\sqrt{\frac{96{x}^{7}}{121}}$	70. $\sqrt{\frac{108{y}^{4}}{49}}$
71. $\sqrt{\frac{300{m}^{5}}{64}}$	72. $\sqrt{\frac{125{n}^{7}}{169}}$
73. $\sqrt{\frac{98{r}^{5}}{100}}$	74. $\sqrt{\frac{180{s}^{10}}{144}}$
75. $\sqrt{\frac{28{q}^{6}}{225}}$	76. $\sqrt{\frac{150{r}^{3}}{256}}$
77. $\sqrt{\frac{75{r}^{9}}{{s}^{8}}}$	78. $\sqrt{\frac{72{x}^{5}}{{y}^{6}}}$
79. $\sqrt{\frac{28{p}^{7}}{{q}^{2}}}$	80. $\sqrt{\frac{45{r}^{3}}{{s}^{10}}}$
81. $\sqrt{\frac{100{x}^{5}}{36{x}^{3}}}$	82. $\sqrt{\frac{49{r}^{12}}{16{r}^{6}}}$
83. $\sqrt{\frac{121{p}^{5}}{81{p}^{2}}}$	84. $\sqrt{\frac{25{r}^{8}}{64r}}$
85. $\sqrt{\frac{32{x}^{5}{y}^{3}}{18{x}^{3}y}}$	86. $\sqrt{\frac{75{r}^{6}{s}^{8}}{48r{s}^{4}}}$
87. $\sqrt{\frac{27{p}^{2}q}{108{p}^{5}{q}^{3}}}$	88. $\sqrt{\frac{50{r}^{5}{s}^{2}}{128{r}^{2}{s}^{5}}}$

1. $3\sqrt{3}$	3. $5\sqrt{5}$	5. $10\sqrt{2}$
7. $15\sqrt{2}$	9. $20\sqrt{2}$	11. $15\sqrt{3}$
13. ${x}^{3}\sqrt{x}$	15. $p\sqrt{p}$	17. ${m}^{6}\sqrt{m}$
19. ${r}^{12}\sqrt{r}$	21. $7{n}^{8}\sqrt{n}$	23. $9{r}^{7}\sqrt{r}$
25. $7{m}^{2}\sqrt{2m}$	27. $5{r}^{6}\sqrt{5r}$	29. $10{p}^{6}\sqrt{2p}$
31. $11{m}^{11}\sqrt{2m}$	33. $7{m}^{3}{n}^{5}\sqrt{3mn}$	35. $5{r}^{6}{s}^{4}\sqrt{3rs}$ 70)
37. $10{p}^{4}{q}^{5}\sqrt{3pq}$	39. $11{m}^{6}{n}^{10}\sqrt{2mn}$	41. $5+2\sqrt{3}$
43. $1+3\sqrt{5}$	45. $5-2\sqrt{6}$	47. $1+\sqrt{10}$
49. $\frac{7}{8}$	51. $\frac{11}{4}$	53. $\frac{6}{7}$
55. $\frac{3}{5}$	57. ${x}^{2}$	59. $\frac{1}{{y}^{2}}$
61. $10{x}^{2}$	63. $4{p}^{4}$	65. $\frac{6}{\sqrt{35}}$
67. $\frac{2\sqrt{5}}{9}$	69. $\frac{4{x}^{3}\sqrt{6x}}{11}$	71. $\frac{10{m}^{2}\sqrt{3m}}{8}$
73. $\frac{7{r}^{2}\sqrt{2r}}{10}$	75. $\frac{2{q}^{3}\sqrt{7}}{15}$	77. $\frac{5{r}^{4}\sqrt{3r}}{{s}^{4}}$
79. $\frac{4{p}^{3}\sqrt{7p}}{q}$	81. $\frac{5x}{3}$	83. $\frac{11p\sqrt{p}}{9}$
85. $\frac{4xy}{3}$	87. $\frac{1}{2pq\sqrt{p}}$	89. a) $17.0\phantom{\rule{0.2em}{0ex}}\text{feet}$ b) $15.0\phantom{\rule{0.2em}{0ex}}\text{feet}$
91. Answers will vary.

7.5 Simplify Square Roots

Use the Product Property to Simplify Square Roots

Use the Quotient Property to Simplify Square Roots

Everyday Math

Writing Exercises

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Use the Product Property to Simplify Square Roots

Use the Quotient Property to Simplify Square Roots

Key Concepts

Practice Makes Perfect

Use the Product Property to Simplify Square Roots

Use the Quotient Property to Simplify Square Roots

Everyday Math

Writing Exercises

Answers:

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