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 Figure 1 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.

So $\sqrt{31}$ is simplified. But $\sqrt{32}$ 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 ${(a b)}^{m} = a^{m} b^{m}$ . 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.

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.

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.

Simplify: $\sqrt{500}$ .

Solution

Illustrates the step-by-step process of simplifying the square root of 500 by factoring out perfect squares.
	$\sqrt{500}$
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.	$10 \sqrt{5}$

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.

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

Solution

This table illustrates the step-by-step simplification of the square root expression sqrt(x^3), showing the procedure and the evolving mathematical form.
	$\sqrt{x^{3}}$
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.	$x \sqrt{x}$

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

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

Solution

Illustrates the step-by-step process of simplifying the square root expression sqrt(25y^5).
	$\sqrt{25 y^{5}}$
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.	$5 y^{2} \sqrt{y}$

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

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

Solution

Step-by-step simplification of a square root expression (sqrt(72n^7)), showing the process of extracting perfect square factors.
	$\sqrt{72 n^{7}}$
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.	$6 n^{3} \sqrt{2 n}$

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

Solution

This table illustrates the step-by-step process of simplifying the radical expression sqrt(63u^3v^5).
	$\sqrt{63 u^{3} v^{5}}$
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.	$3 u v^{2} \sqrt{7 u v}$

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.

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

Solution

This table demonstrates the step-by-step simplification of the radical expression 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.

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

Solution

Step-by-step simplification of the radical expression (4 - 48)/2 to its simplified form 2(1 - 3).
	$\frac{4 - \sqrt{48}}{2}$
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.

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

Solution

$\begin{matrix} \sqrt{\frac{9}{64}} \\ Since {(\frac{3}{8})}^{2} = \frac{9}{64} & \frac{3}{8} \end{matrix}$

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

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

Solution

This table demonstrates the step-by-step process of simplifying a square root of a fraction, showing both the instructions and the corresponding mathematical expressions.
	$\sqrt{\frac{45}{80}}$
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, $\frac{a^{m}}{a^{n}} = a^{m - n}, a \neq 0$ .

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

Solution

Step-by-step simplification of the radical expression involving variables and exponents.
	$\sqrt{\frac{m^{6}}{m^{4}}}$
Simplify the fraction inside the radical first.
Divide the like bases by subtracting the exponents.	$\sqrt{m^{2}}$
Simplify.	$m$

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

Solution

This table illustrates the step-by-step simplification of a radical algebraic expression.
	$\sqrt{\frac{48 p^{7}}{3 p^{3}}}$
Simplify the fraction inside the radical first.	$\sqrt{16 p^{4}}$
Simplify.	$4 p^{2}$

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.

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

Solution

Step-by-step simplification of the radical expression sqrt(21/64) using the quotient property of square roots.
	$\sqrt{\frac{21}{64}}$
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.	$\frac{\sqrt{21}}{8}$

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 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.

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

Solution

Step-by-step simplification of a radical expression involving a fraction, illustrating the application of the Quotient Property of radicals.
	$\sqrt{\frac{45 x^{5}}{y^{4}}}$
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.

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

Solution

Step-by-step simplification of a radical expression involving variables and fractions.
	$\sqrt{\frac{81 d^{9}}{25 d^{4}}}$
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.	$\frac{9 d^{2} \sqrt{d}}{5}$

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

Solution

Step-by-step simplification of a radical expression with fractions and variables using algebraic properties.
	$\sqrt{\frac{18 p^{5} q^{7}}{32 p q^{2}}}$
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 $b \neq 0$ , 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.

$\sqrt{27}$

Solution

$3 \sqrt{3}$

$\sqrt{80}$

$\sqrt{125}$

Solution

$5 \sqrt{5}$

$\sqrt{96}$

$\sqrt{200}$

Solution

$10 \sqrt{2}$

$\sqrt{147}$

$\sqrt{450}$

Solution

$15 \sqrt{2}$

$\sqrt{252}$

$\sqrt{800}$

Solution

$20 \sqrt{2}$

$\sqrt{288}$

$\sqrt{675}$

Solution

$15 \sqrt{3}$

$\sqrt{1250}$

$\sqrt{x^{7}}$

Solution

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

$\sqrt{y^{11}}$

$\sqrt{p^{3}}$

Solution

$p \sqrt{p}$

$\sqrt{q^{5}}$

$\sqrt{m^{13}}$

Solution

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

$\sqrt{n^{21}}$

$\sqrt{r^{25}}$

Solution

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

$\sqrt{s^{33}}$

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

Solution

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

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

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

Solution

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

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

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

Solution

$7 m^{2} \sqrt{2 m}$

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

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

Solution

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

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

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

Solution

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

$\sqrt{128 q^{3}}$

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

Solution

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

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

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

Solution

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

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

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

Solution

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

$\sqrt{96 r^{3} s^{3}}$

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

Solution

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

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

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

Solution

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

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

$5 + \sqrt{12}$

Solution

$5 + 2 \sqrt{3}$

$8 + \sqrt{96}$

$1 + \sqrt{45}$

Solution

$1 + 3 \sqrt{5}$

$3 + \sqrt{125}$

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

Solution

$5 - \sqrt{6}$

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

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

Solution

$1 + \sqrt{10}$

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

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

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

Solution

$\frac{7}{8}$

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

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

Solution

$\frac{11}{4}$

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

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

Solution

$\frac{6}{7}$

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

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

Solution

$\frac{3}{5}$

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

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

Solution

$x^{2}$

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

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

Solution

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

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

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

Solution

$10 x^{2}$

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

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

Solution

$4 p^{4}$

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

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

Solution

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

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

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

Solution

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

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

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

Solution

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

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

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

Solution

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

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

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

Solution

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

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

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

Solution

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

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

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

Solution

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

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

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

Solution

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

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

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

Solution

$\frac{5 x}{3}$

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

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

Solution

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

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

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

Solution

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

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

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

Solution

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

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

Everyday Math

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

Solution

ⓐ $17.0 feet$ ⓑ $12.0 feet$

ⓐ 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 $\sqrt{x^{16}} = x^{8}$ .

Solution

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?