9.2 Simplify Square Roots

In the last section, we estimated the square root of a number between two consecutive whole numbers. We can say that $the square root of 50$ is between 7 and 8. This is fairly easy to do when the numbers are small enough that we can use Figure 9.2.

But what if we want to estimate $the square root of 500$ ? 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.

So $the square root of 31$ is simplified. But $the square root of 32$ 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 ${(a b)}^{m} = a^{m} b^{m}$ . The corresponding property of square roots says that $\sqrt{a b} = \sqrt{a} \cdot \sqrt{b}$ .

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.

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

We could use the simplified form $10 the square root of 5$ to estimate $the square root of 500$ . We know 5 is between 2 and 3, and $the square root of 500$ is $10 the square root of 5$ . So $the square root of 500$ is between 20 and 30.

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

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

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

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: $3 + \sqrt{32}$ .

	$3 + \sqrt{32}$
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!

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

	$\frac{4 - \sqrt{48}}{2}$
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.	$\frac{4 (1 - \sqrt{3})}{2}$
Remove the common factor, 2, from the numerator and denominator.	$\frac{2 \cdot 2 (1 - \sqrt{3})}{2}$
Simplify.	$2 (1 - \sqrt{3})$

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

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

Example 10

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

	$\sqrt{\frac{45}{80}}$
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.	$\sqrt{\frac{9}{16}}$
$Simplify. {(\frac{3}{4})}^{2} = \frac{9}{16}$	$\frac{3}{4}$

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

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.

${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}, b \neq 0$

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.

Example 15

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

	$\sqrt{\frac{45 x^{5}}{y^{4}}}$
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.	$\frac{\sqrt{9 x^{4}} \cdot \sqrt{5 x}}{y^{2}}$
Simplify.	$\frac{3 x^{2} \sqrt{5 x}}{y^{2}}$

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

Example 16

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

	$\sqrt{\frac{81 d^{9}}{25 d^{4}}}$
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.	$\frac{9 d^{2} \sqrt{d}}{5}$

Example 17

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

	$\sqrt{\frac{18 p^{5} q^{7}}{32 p q^{2}}}$
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.	$\frac{3 p^{2} q^{2} \sqrt{q}}{4}$

Key Concepts

Simplified Square Root $\sqrt{a}$ is considered simplified if $a$ has no perfect-square factors.
Product Property of Square Roots If a, b are non-negative real numbers, then

$\sqrt{a b} = \sqrt{a} \cdot \sqrt{b}$
Simplify a Square Root Using the Product Property To simplify a square root using the Product Property:
1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect square factor.
2. Use the product rule to rewrite the radical as the product of two radicals.
3. Simplify the square root of the perfect square.
Quotient Property of Square Roots If a, b are non-negative real numbers and $b not equal to 0$ , then

$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$
Simplify a Square Root Using the Quotient Property To simplify a square root using the Quotient Property:
1. Simplify the fraction in the radicand, if possible.
2. Use the Quotient Rule to rewrite the radical as the quotient of two radicals.
3. Simplify the radicals in the numerator and the denominator.

Practice Makes Perfect

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$the square root of 27$

$3 the square root of 3$

$the square root of 80$

$the square root of 125$

$5 the square root of 5$

$the square root of 96$

$the square root of 200$

$10 the square root of 2$

$the square root of 147$

$the square root of 450$

$15 the square root of 2$

$the square root of 252$

$the square root of 800$

$20 the square root of 2$

$the square root of 288$

$the square root of 675$

$15 the square root of 3$

$the square root of 1250$

$\sqrt{x^{7}}$

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

$\sqrt{y^{11}}$

$\sqrt{p^{3}}$

$p \sqrt{p}$

$\sqrt{q^{5}}$

$\sqrt{m^{13}}$

$m^{6} \sqrt{m}$

$\sqrt{n^{21}}$

$\sqrt{r^{25}}$

$r^{12} \sqrt{r}$

$\sqrt{s^{33}}$

$\sqrt{49 n^{17}}$

$7 n^{8} \sqrt{n}$

$\sqrt{25 m^{9}}$

$\sqrt{81 r^{15}}$

$9 r^{7} \sqrt{r}$

$\sqrt{100 s^{19}}$

$\sqrt{98 m^{5}}$

$7 m squared the square root of 2 m$

$\sqrt{32 n^{11}}$

$\sqrt{125 r^{13}}$

$5 r^{6} \sqrt{5 r}$

$\sqrt{80 s^{15}}$

$\sqrt{200 p^{13}}$

$10 p^{6} \sqrt{2 p}$

$the square root of 128 q cubed$

$\sqrt{242 m^{23}}$

$11 m^{11} \sqrt{2 m}$

$\sqrt{175 n^{13}}$

$\sqrt{147 m^{7} n^{11}}$

$7 m^{3} n^{5} \sqrt{3 m n}$

$\sqrt{48 m^{7} n^{5}}$

$\sqrt{75 r^{13} s^{9}}$

$5 r^{6} s^{4} \sqrt{3 r s}$

$the square root of 96 r cubed s cubed$

$\sqrt{300 p^{9} q^{11}}$

$10 p^{4} q^{5} \sqrt{3 p q}$

$\sqrt{192 q^{3} r^{7}}$

$\sqrt{242 m^{13} n^{21}}$

$11 m^{6} n^{10} \sqrt{2 m n}$

$\sqrt{150 m^{9} n^{3}}$

$5 + \sqrt{12}$

$5 + 2 \sqrt{3}$

$8 + \sqrt{96}$

$1 + \sqrt{45}$

$1 + 3 \sqrt{5}$

$3 + \sqrt{125}$

$\frac{10 - \sqrt{24}}{2}$

$5 - \sqrt{6}$

$\frac{8 - \sqrt{80}}{4}$

$\frac{3 + \sqrt{90}}{3}$

$1 + \sqrt{10}$

$\frac{15 + \sqrt{75}}{5}$

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{49}{64}}$

$\frac{7}{8}$

$\sqrt{\frac{100}{36}}$

$\sqrt{\frac{121}{16}}$

$\frac{11}{4}$

$\sqrt{\frac{144}{169}}$

$\sqrt{\frac{72}{98}}$

$\frac{6}{7}$

$\sqrt{\frac{75}{12}}$

$\sqrt{\frac{45}{125}}$

$\frac{3}{5}$

$\sqrt{\frac{300}{243}}$

$\sqrt{\frac{x^{10}}{x^{6}}}$

$x^{2}$

$\sqrt{\frac{p^{20}}{p^{10}}}$

$\sqrt{\frac{y^{4}}{y^{8}}}$

$\frac{1}{y^{2}}$

$\sqrt{\frac{q^{8}}{q^{14}}}$

$\sqrt{\frac{200 x^{7}}{2 x^{3}}}$

$10 x squared$

$\sqrt{\frac{98 y^{11}}{2 y^{5}}}$

$\sqrt{\frac{96 p^{9}}{6 p}}$

$4 p to the power 4$

$\sqrt{\frac{108 q^{10}}{3 q^{2}}}$

$\sqrt{\frac{36}{35}}$

$\frac{6}{\sqrt{35}}$

$\sqrt{\frac{144}{65}}$

$\sqrt{\frac{20}{81}}$

$\frac{2 \sqrt{5}}{9}$

$\sqrt{\frac{21}{196}}$

$\sqrt{\frac{96 x^{7}}{121}}$

$\frac{4 x^{3} \sqrt{6 x}}{11}$

$\sqrt{\frac{108 y^{4}}{49}}$

$\sqrt{\frac{300 m^{5}}{64}}$

$\frac{5 m^{2} \sqrt{3 m}}{4}$

$\sqrt{\frac{125 n^{7}}{169}}$

$\sqrt{\frac{98 r^{5}}{100}}$

$\frac{7 r^{2} \sqrt{2 r}}{10}$

$\sqrt{\frac{180 s^{10}}{144}}$

$\sqrt{\frac{28 q^{6}}{225}}$

$\frac{2 q^{3} \sqrt{7}}{15}$

$\sqrt{\frac{150 r^{3}}{256}}$

$\sqrt{\frac{75 r^{9}}{s^{8}}}$

$\frac{5 r^{4} \sqrt{3 r}}{s^{4}}$

$\sqrt{\frac{72 x^{5}}{y^{6}}}$

$\sqrt{\frac{28 p^{7}}{q^{2}}}$

$\frac{2 p^{3} \sqrt{7 p}}{q}$

$\sqrt{\frac{45 r^{3}}{s^{10}}}$

$\sqrt{\frac{100 x^{5}}{36 x^{3}}}$

$\frac{5 x}{3}$

$\sqrt{\frac{49 r^{12}}{16 r^{6}}}$

$\sqrt{\frac{121 p^{5}}{81 p^{2}}}$

$\frac{11 p \sqrt{p}}{9}$

$\sqrt{\frac{25 r^{8}}{64 r}}$

$\sqrt{\frac{32 x^{5} y^{3}}{18 x^{3} y}}$

$\frac{4 x y}{3}$

$\sqrt{\frac{75 r^{6} s^{8}}{48 r s^{4}}}$

$\sqrt{\frac{27 p^{2} q}{108 p^{5} q^{3}}}$

$\frac{1}{2 p q \sqrt{p}}$

$\sqrt{\frac{50 r^{5} s^{2}}{128 r^{2} s^{5}}}$

Everyday Math

ⓐ Elliott decides to construct a square garden that will take up 288 square feet of his yard. Simplify $the square root of 288$ 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 $\sqrt{288} - 5$ to determine the length and width of the new garden. Round to the nearest tenth of a foot.

ⓐ $17.0 feet$ ⓑ $12.0 feet$

ⓐ Melissa accidentally drops a pair of sunglasses from the top of a roller coaster, 64 feet above the ground. Simplify $\sqrt{\frac{64}{16}}$ 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 $\sqrt{\frac{144}{16}}$ to determine the number of seconds it takes for the sunglasses to reach the ground.

Writing Exercises

Explain why $\sqrt{x^{4}} = x^{2}$ . Then explain why $\sqrt{x^{16}} = x^{8}$ .

Answers will vary.

Explain why $7 + \sqrt{9}$ is not equal to $\sqrt{7 + 9}$ .

Self Check

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

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.

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

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

	$\sqrt{25 y^{5}}$
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.	$5 y squared the square root of y$

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

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

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

	$the square root of 500$
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.	$10 the square root of 5$

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

	$\sqrt{\frac{21}{64}}$
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.	$\frac{\sqrt{21}}{8}$