Simplify Radical Expressions - Intermediate Algebra

8.2 Simplify Radical Expressions

Learning Objectives

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

Use the Product Property to simplify radical expressions
Use the Quotient Property to simplify radical expressions

Be Prepared 8.4

Before you get started, take this readiness quiz.

Simplify: $\frac{x^{9}}{x^{4}} .$
If you missed this problem, review Example 5.13.

Be Prepared 8.5

Simplify: $\frac{y^{3}}{y^{11}} .$
If you missed this problem, review Example 5.13.

Be Prepared 8.6

Simplify: ${(n^{2})}^{6} .$
If you missed this problem, review Example 5.17.

Use the Product Property to Simplify Radical Expressions

We will simplify radical expressions in a way similar to how we simplified fractions. A fraction is simplified if there are no common factors in the numerator and denominator. To simplify a fraction, we look for any common factors in the numerator and denominator.

A radical expression, $\sqrt[n]{a},$ is considered simplified if it has no factors of $m^{n} .$ So, to simplify a radical expression, we look for any factors in the radicand that are powers of the index.

Simplified Radical Expression

For real integers or rational numbers a and m, and $n \geq 2,$

\sqrt[n]{a} is considered simplified if a has no factors of m^{n}

For example, $\sqrt{5}$ is considered simplified because there are no perfect square factors in 5. But $\sqrt{12}$ is not simplified because 12 has a perfect square factor of 4.

Similarly, $\sqrt[3]{4}$ is simplified because there are no perfect cube factors in 4. But $\sqrt[3]{24}$ is not simplified because 24 has a perfect cube factor of 8.

To simplify radical expressions, we will also use some properties of roots. The properties we will use to simplify radical expressions are similar to the properties of exponents. We know that ${(a b)}^{n} = a^{n} b^{n} .$ The corresponding of Product Property of Roots says that $\sqrt[n]{a b} = \sqrt[n]{a} \cdot \sqrt[n]{b} .$

Product Property of n^th Roots

If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, and $n \geq 2$ is an integer, then

\sqrt[n]{a b} = \sqrt[n]{a} \cdot \sqrt[n]{b} and \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a b}

We use the Product Property of Roots to remove all perfect square factors from a square root.

Example 8.13

Simplify Square Roots Using the Product Property of Roots

Simplify: $\sqrt{98} .$

Solution

The first step in the process is to find the largest factor in the radicand that is a perfect power of the index and rewrite the radicand as a product of two factors, using that factor. We see that 49 is the largest factor of 98 that has a power of 2. In other words 49 is the largest perfect square factor of 98. We can write 98 equals 49 times 2. Always write the perfect square factor first. The square root of 98 can then be written as the square root of the quantity 49 times 2 in parentheses.

The second step in the process is to use the product rule to rewrite the radical as the product of two radicals. The square root of the quantity 49 times 2 in parentheses can be written as the square root of 49 times the square root of 2.

The third step is to simplify the root of the perfect power. The square root of 49 times the square root of 2 can be written as 7 times the square root of 2.

Try It 8.25

Simplify: $\sqrt{48} .$

Try It 8.26

Simplify: $\sqrt{45} .$

Notice in the previous example that the simplified form of $\sqrt{98}$ is $7 \sqrt{2},$ which is the product of an integer and a square root. We always write the integer in front of the square root.

Be careful to write your integer so that it is not confused with the index. The expression $7 \sqrt{2}$ is very different from $\sqrt[7]{2} .$

How To

Simplify a radical expression using the Product Property.

Step 1. Find the largest factor in the radicand that is a perfect power of the index. Rewrite the radicand as a product of two factors, using that factor.
Step 2. Use the product rule to rewrite the radical as the product of two radicals.
Step 3. Simplify the root of the perfect power.

We will apply this method in the next example. It may be helpful to have a table of perfect squares, cubes, and fourth powers.

Example 8.14

Simplify: ⓐ $\sqrt{500}$ ⓑ $\sqrt[3]{16}$ ⓒ $\sqrt[4]{243} .$

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.	$10 \sqrt{5}$

Steps demonstrating how to simplify the square root of 500 into its simplest radical form.

ⓑ

	$\sqrt[3]{16}$
Rewrite the radicand as a product using the greatest perfect cube factor. $2^{3} = 8$	$\sqrt[3]{8 \cdot 2}$
Rewrite the radical as the product of two radicals.	$\sqrt[3]{8} \cdot \sqrt[3]{2^{}}$
Simplify.	$2 \sqrt[3]{2^{}}$

This table demonstrates the step-by-step simplification of the cube root of 16.

ⓒ

	$\sqrt[4]{243}$
Rewrite the radicand as a product using the greatest perfect fourth power factor. $3^{4} = 81$	$\sqrt[4]{81 \cdot 3^{}}$
Rewrite the radical as the product of two radicals.	$\sqrt[4]{81} \cdot \sqrt[4]{3^{}}$
Simplify.	$3 \sqrt[4]{3^{}}$

Step-by-step simplification of the fourth root of 243, illustrating the process of extracting perfect fourth power factors.

Try It 8.27

Simplify: ⓐ $\sqrt{288}$ ⓑ $\sqrt[3]{81}$ ⓒ $\sqrt[4]{64} .$

Try It 8.28

Simplify: ⓐ $\sqrt{432}$ ⓑ $\sqrt[3]{625}$ ⓒ $\sqrt[4]{729} .$

The next example is much like the previous examples, but with variables. Don’t forget to use the absolute value signs when taking an even root of an expression with a variable in the radical.

Example 8.15

Simplify: ⓐ $\sqrt{x^{3}}$ ⓑ $\sqrt[3]{x^{4}}$ ⓒ $\sqrt[4]{x^{7}} .$

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.	$\| x \| \sqrt{x}$

Step-by-step simplification of the radical expression sqrt(x^3).

ⓑ

	$\sqrt[3]{x^{4}}$
Rewrite the radicand as a product using the largest perfect cube factor.	$\sqrt[3]{x^{3} \cdot x} .$
Rewrite the radical as the product of two radicals.	$\sqrt[3]{x^{3}} \cdot \sqrt[3]{x}$
Simplify.	$x \sqrt[3]{x}$

Step-by-step simplification of the radical expression abdul,{

ⓒ

	$\sqrt[4]{x^{7}}$
Rewrite the radicand as a product using the greatest perfect fourth power factor.	$\sqrt[4]{x^{4} \cdot x^{3}}$
Rewrite the radical as the product of two radicals.	$\sqrt[4]{x^{4}} \cdot \sqrt[4]{x^{3}}$
Simplify.	$\| x \| \sqrt[4]{x^{3}}$

This table illustrates the step-by-step process of simplifying a radical expression.

Try It 8.29

Simplify: ⓐ $\sqrt{b^{5}}$ ⓑ $\sqrt[4]{y^{6}}$ ⓒ $\sqrt[3]{z^{5}}$

Try It 8.30

Simplify: ⓐ $\sqrt{p^{9}}$ ⓑ $\sqrt[5]{y^{8}}$ ⓒ $\sqrt[6]{q^{13}}$

We follow the same procedure when there is a coefficient in the radicand. In the next example, both the constant and the variable have perfect square factors.

Example 8.16

Simplify: ⓐ $\sqrt{72 n^{7}}$ ⓑ $\sqrt[3]{24 x^{7}}$ ⓒ $\sqrt[4]{80 y^{14}} .$

Solution

ⓐ

	$\sqrt{72 n^{7}}$
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.	$6 \| n^{3} \| \sqrt{2 n}$

Illustrates the step-by-step process of simplifying the radical expression sqrt(72n^7) to 6|n^3|sqrt(2n).

ⓑ

	$\sqrt[3]{24 x^{7}}$
Rewrite the radicand as a product using perfect cube factors.	$\sqrt[3]{8 x^{6} \cdot 3 x}$
Rewrite the radical as the product of two radicals.	$\sqrt[3]{8 x^{6}} \cdot \sqrt[3]{3^{} x}$
Rewrite the first radicand as ${(2 x^{2})}^{3} .$	$\sqrt[3]{{(2^{} x^{2})}^{3}} \cdot \sqrt[3]{3^{} x}$
Simplify.	$2 x^{2} \sqrt[3]{3^{} x}$

Step-by-step process for simplifying the cube root of 24x^7.

ⓒ

	$\sqrt[4]{80 y^{14}}$
Rewrite the radicand as a product using perfect fourth power factors.	$\sqrt[4]{16 y^{12} \cdot 5^{} y^{2}}$
Rewrite the radical as the product of two radicals.	$\sqrt[4]{16 y^{12}} \cdot \sqrt[4]{5 y^{2}^{}}$
Rewrite the first radicand as ${(2 y^{3})}^{4} .$	$\sqrt[4]{{(2^{} y^{3})}^{4}} \cdot \sqrt[4]{5 y^{2}^{}}$
Simplify.	$2 \| y^{3} \| \sqrt[4]{5^{} y^{2}}$

Step-by-step process demonstrating the simplification of the fourth root of 80y^14.

Try It 8.31

Simplify: ⓐ $\sqrt{32 y^{5}}$ ⓑ $\sqrt[3]{54 p^{10}}$ ⓒ $\sqrt[4]{64 q^{10}} .$

Try It 8.32

Simplify: ⓐ $\sqrt{75 a^{9}}$ ⓑ $\sqrt[3]{128 m^{11}}$ ⓒ $\sqrt[4]{162 n^{7}} .$

In the next example, we continue to use the same methods even though there are more than one variable under the radical.

Example 8.17

Simplify: ⓐ $\sqrt{63 u^{3} v^{5}}$ ⓑ $\sqrt[3]{40 x^{4} y^{5}}$ ⓒ $\sqrt[4]{48 x^{4} y^{7}} .$

Solution

ⓐ

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

Step-by-step process demonstrating the simplification of a complex square root expression involving variables.

ⓑ

	$\sqrt[3]{40 x^{4} y^{5}}$
Rewrite the radicand as a product using the largest perfect cube factor.	$\sqrt[3]{8 x^{3} y^{3} \cdot 5 x y^{2}}$
Rewrite the radical as the product of two radicals.	$\sqrt[3]{8 x^{3} y^{3}} \cdot \sqrt[3]{5 x y^{2}}$
Rewrite the first radicand as ${(2 x y)}^{3} .$	$\sqrt[3]{{(2 x y)}^{3}} \cdot \sqrt[3]{5 x y^{2}}$
Simplify.	$2 x y \sqrt[3]{5 x y^{2}}$

This table demonstrates the step-by-step process of simplifying a cube root expression, showing each operation and its corresponding mathematical form.

ⓒ

	$\sqrt[4]{48 x^{4} y^{7}}$
Rewrite the radicand as a product using the largest perfect fourth power factor.	$\sqrt[4]{16 x^{4} y^{4} \cdot 3 y^{3}}$
Rewrite the radical as the product of two radicals.	$\sqrt[4]{16 x^{4} y^{4}} \cdot \sqrt[4]{3 y^{3}}$
Rewrite the first radicand as ${(2 x y)}^{4} .$	$\sqrt[4]{{(2 x y)}^{4}} \cdot \sqrt[4]{3 y^{3}}$
Simplify.	$2 \| x y \| \sqrt[4]{3 y^{3}}$

Step-by-step simplification of the radical expression fourth root of 48x^4y^7, demonstrating each algebraic transformation.

Try It 8.33

Simplify: ⓐ $\sqrt{98 a^{7} b^{5}}$ ⓑ $\sqrt[3]{56 x^{5} y^{4}}$ ⓒ $\sqrt[4]{32 x^{5} y^{8}} .$

Try It 8.34

Simplify: ⓐ $\sqrt{180 m^{9} n^{11}}$ ⓑ $\sqrt[3]{72 x^{6} y^{5}}$ ⓒ $\sqrt[4]{80 x^{7} y^{4}} .$

Example 8.18

Simplify: ⓐ $\sqrt[3]{−27}$ ⓑ $\sqrt[4]{−16} .$

Solution

ⓐ

	$\sqrt[3]{−27}$
Rewrite the radicand as a product using perfect cube factors.	$\sqrt[3]{{(−3)}^{3}}$
Take the cube root.	$−3$

Step-by-step demonstration of calculating the cube root of -27.

ⓑ

	$\sqrt[4]{−16}$
There is no real number $n$ where $n^{4} = −16 .$	Not a real number.

Illustrates why the fourth root of a negative number, such as ⁴√(-16), is not a real number.

Try It 8.35

Simplify: ⓐ $\sqrt[3]{−64}$ ⓑ $\sqrt[4]{−81} .$

Try It 8.36

Simplify: ⓐ $\sqrt[3]{−625}$ ⓑ $\sqrt[4]{−324} .$

We have seen how to use the order of operations to simplify some expressions with radicals. 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 since one term contains a radical and the other does not. The next example also 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 8.19

Simplify: ⓐ $3 + \sqrt{32}$ ⓑ $\frac{4 - \sqrt{48}}{2} .$

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.	$3 + 4 \sqrt{2}$

This table illustrates the step-by-step process for simplifying the mathematical expression 3 + sqrt(32).

The terms cannot be added as one has a radical and the other does not. Trying to add an integer and a radical is like trying to add an integer and a variable. They are not like terms!

ⓑ

	$\frac{4 - \sqrt{48}}{2}$
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.	$2 (1 - \sqrt{3})$

Step-by-step simplification of the mathematical expression (4 - sqrt(48))/2 to its simplified form 2(1 - sqrt(3)).

Try It 8.37

Simplify: ⓐ $5 + \sqrt{75}$ ⓑ $\frac{10 - \sqrt{75}}{5}$

Try It 8.38

Simplify: ⓐ $2 + \sqrt{98}$ ⓑ $\frac{6 - \sqrt{45}}{3}$

Use the Quotient Property to Simplify Radical Expressions

Whenever you have to simplify a radical expression, the first step you should take is to determine whether the radicand is a perfect power of the index. If not, check the numerator and denominator for any common factors, and remove them. You may find a fraction in which both the numerator and the denominator are perfect powers of the index.

Example 8.20

Simplify: ⓐ $\sqrt{\frac{45}{80}}$ ⓑ $\sqrt[3]{\frac{16}{54}}$ ⓒ $\sqrt[4]{\frac{5}{80}} .$

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

This table illustrates the step-by-step process of simplifying the square root of a fraction, starting from sqrt(45/80) and ending with 3/4.

ⓑ

	$\sqrt[3]{\frac{16}{54}}$
Simplify inside the radical first.
Rewrite showing the common factors of the numerator and denominator.	$\sqrt[3]{\frac{2 \cdot 8}{2 \cdot 27}}$
Simplify the fraction by removing common factors.	$\sqrt[3]{\frac{8}{27}}$
Simplify. Note ${(\frac{2}{3})}^{3} = \frac{8}{27} .$	$\frac{2}{3}$

Step-by-step simplification of the cube root of a fraction by simplifying the expression inside the radical.

ⓒ

	$\sqrt[4]{\frac{5}{80}}$
Simplify inside the radical first.
Rewrite showing the common factors of the numerator and denominator.	$\sqrt[4]{\frac{5 \cdot 1}{5 \cdot 16}}$
Simplify the fraction by removing common factors.	$\sqrt[4]{\frac{1}{16}}$
Simplify. Note ${(\frac{1}{2})}^{4} = \frac{1}{16} .$	$\frac{1}{2}$

This table illustrates the step-by-step process for simplifying the fourth root of 5/80, culminating in the simplified value of 1/2.

Try It 8.39

Simplify: ⓐ $\sqrt{\frac{75}{48}}$ ⓑ $\sqrt[3]{\frac{54}{250}}$ ⓒ $\sqrt[4]{\frac{32}{162}} .$

Try It 8.40

Simplify: ⓐ $\sqrt{\frac{98}{162}}$ ⓑ $\sqrt[3]{\frac{24}{375}}$ ⓒ $\sqrt[4]{\frac{4}{324}} .$

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

Example 8.21

Simplify: ⓐ $\sqrt{\frac{m^{6}}{m^{4}}}$ ⓑ $\sqrt[3]{\frac{a^{8}}{a^{5}}}$ ⓒ $\sqrt[4]{\frac{a^{10}}{a^{2}}} .$

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.	$\| m \|$

Step-by-step simplification of a radical expression involving variables with exponents.

ⓑ

	$\sqrt[3]{\frac{a^{8}}{a^{5}}}$
Use the Quotient Property of exponents to simplify the fraction under the radical first.	$\sqrt[3]{a^{3}}$
Simplify.	$a$

Steps to simplify a cube root expression using the Quotient Property of exponents.

ⓒ

	$\sqrt[4]{\frac{a^{10}}{a^{2}}}$
Use the Quotient Property of exponents to simplify the fraction under the radical first.	$\sqrt[4]{a^{8}}$
Rewrite the radicand using perfect fourth power factors.	$\sqrt[4]{{(a^{2})}^{4}}$
Simplify.	$a^{2}$

Step-by-step guide on simplifying a radical expression involving variable exponents, illustrating the application of the Quotient Property.

Try It 8.41

Simplify: ⓐ $\sqrt{\frac{a^{8}}{a^{6}}}$ ⓑ $\sqrt[4]{\frac{x^{7}}{x^{3}}}$ ⓒ $\sqrt[4]{\frac{y^{17}}{y^{5}}} .$

Try It 8.42

Simplify: ⓐ $\sqrt{\frac{x^{14}}{x^{10}}}$ ⓑ $\sqrt[3]{\frac{m^{13}}{m^{7}}}$ ⓒ $\sqrt[5]{\frac{n^{12}}{n^{2}}} .$

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 root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect power of the index, we simplify the numerator and denominator separately.

Quotient Property of Radical Expressions

If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, $b \neq 0,$ and for any integer $n \geq 2$ then,

\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} and \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}

Example 8.22

How to Simplify the Quotient of Radical Expressions

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

Solution

$The first step in the process is to simplify the fraction in the radicand, if possible. In this example the quantity 27 m cubed in parentheses divided by 196 cannot be simplified.$ The second step in the process is to use the quotient property to rewrite the radical as the quotient of two radicals. We rewrite the square root of the quantity 27 m cubed divided by 196 in parentheses as the quotient of the square root of the quantity 27 m cubed in parentheses and the square root of 196.

The second step in the process is to use the quotient property to rewrite the radical as the quotient of two radicals. We rewrite the square root of the quantity 27 m cubed divided by 196 in parentheses as the quotient of the square root of the quantity 27 m cubed in parentheses and the square root of 196.

The third step is to simplify the radicals in the numerator and the denominator. 9 m squared and 196 are perfect squares. We rewrite the expression as the quantity square root of quantity 9 m squared in parentheses times square root of the quantity 3 m in parentheses in parentheses divided by square root of 196. The simplified version is the quantity 3 absolute value m times square root of the quantity 3 m in parentheses in parentheses divided by 14.

Try It 8.43

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

Try It 8.44

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

How To

Simplify a square root using the Quotient Property.

Step 1. Simplify the fraction in the radicand, if possible.
Step 2. Use the Quotient Property to rewrite the radical as the quotient of two radicals.
Step 3. Simplify the radicals in the numerator and the denominator.

Example 8.23

Simplify: ⓐ $\sqrt{\frac{45 x^{5}}{y^{4}}}$ ⓑ $\sqrt[3]{\frac{24 x^{7}}{y^{3}}}$ ⓒ $\sqrt[4]{\frac{48 x^{10}}{y^{8}}} .$

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

This table illustrates the step-by-step process of simplifying the radical expression (45x^5/y^4).

ⓑ

	$\sqrt[3]{\frac{24 x^{7}}{y^{3}}}$
The fraction in the radicand cannot be simplified. Use the Quotient Property to write as two radicals.	$\frac{\sqrt[3]{24 x^{7}}}{\sqrt[3]{y^{3}}}$
Rewrite each radicand as a product using perfect cube factors.	$\frac{\sqrt[3]{8 x^{6} \cdot 3 x}}{\sqrt[3]{y^{3}}}$
Rewrite the numerator as the product of two radicals.	$\frac{\sqrt[3]{{(2 x^{2})}^{3}} \cdot \sqrt[3]{3 x}}{\sqrt[3]{y^{3}}}$
Simplify.	$\frac{2 x^{2} \sqrt[3]{3 x}}{y}$

Step-by-step method for simplifying a cube root expression with a fractional radicand, demonstrating the application of radical properties.

ⓒ

	$\sqrt[4]{\frac{48 x^{10}}{y^{8}}}$
The fraction in the radicand cannot be simplified.	$\frac{\sqrt[4]{48 x^{10}}}{\sqrt[4]{y^{8}}}$
Use the Quotient Property to write as two radicals. Rewrite each radicand as a product using perfect fourth power factors.	$\frac{\sqrt[4]{16 x^{8} \cdot 3 x^{2}}}{\sqrt[4]{y^{8}}}$
Rewrite the numerator as the product of two radicals.	$\frac{\sqrt[4]{{(2 x^{2})}^{4}} \cdot \sqrt[4]{3 x^{2}}}{\sqrt[4]{{(y^{2})}^{4}}}$
Simplify.	$\frac{2 x^{2} \sqrt[4]{3 x^{2}}}{y^{2}}$

This table illustrates the step-by-step process of simplifying a fourth root expression containing a fraction with variables, utilizing properties of radicals.

Try It 8.45

Simplify: ⓐ $\sqrt{\frac{80 m^{3}}{n^{6}}}$ ⓑ $\sqrt[3]{\frac{108 c^{10}}{d^{6}}}$ ⓒ $\sqrt[4]{\frac{80 x^{10}}{y^{4}}} .$

Try It 8.46

Simplify: ⓐ $\sqrt{\frac{54 u^{7}}{v^{8}}}$ ⓑ $\sqrt[3]{\frac{40 r^{3}}{s^{6}}}$ ⓒ $\sqrt[4]{\frac{162 m^{14}}{n^{12}}} .$

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

Example 8.24

Simplify: ⓐ $\sqrt{\frac{18 p^{5} q^{7}}{32 p q^{2}}}$ ⓑ $\sqrt[3]{\frac{16 x^{5} y^{7}}{54 x^{2} y^{2}}}$ ⓒ $\sqrt[4]{\frac{5 a^{8} b^{6}}{80 a^{3} b^{2}}} .$

Solution

ⓐ

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

This table demonstrates the step-by-step process of simplifying a complex square root expression using algebraic rules.

ⓑ

	$\sqrt[3]{\frac{16 x^{5} y^{7}}{54 x^{2} y^{2}}}$
Simplify the fraction in the radicand, if possible.	$\sqrt[3]{\frac{8 x^{3} y^{5}}{27}}$
Rewrite using the Quotient Property.	$\frac{\sqrt[3]{8 x^{3} y^{5}}}{\sqrt[3]{27}}$
Simplify the radicals in the numerator and the denominator.	$\frac{\sqrt[3]{8 x^{3} y^{3}} \cdot \sqrt[3]{y^{2}}}{\sqrt[3]{27}}$
Simplify.	$\frac{2 x y \sqrt[3]{y^{2}}}{3}$

Step-by-step simplification of a cube root expression involving fractions and variables, demonstrating radical properties.

ⓒ

	$\sqrt[4]{\frac{5 a^{8} b^{6}}{80 a^{3} b^{2}}}$
Simplify the fraction in the radicand, if possible.	$\sqrt[4]{\frac{a^{5} b^{4}}{16}}$
Rewrite using the Quotient Property.	$\frac{\sqrt[4]{a^{5} b^{4}}}{\sqrt[4]{16}}$
Simplify the radicals in the numerator and the denominator.	$\frac{\sqrt[4]{a^{4} b^{4}} \cdot \sqrt[4]{a}}{\sqrt[4]{16}}$
Simplify.	$\frac{\| a b \| \sqrt[4]{a}}{2}$

Step-by-step simplification of a fourth root radical expression involving fractions and variables.

Try It 8.47

Simplify: ⓐ $\sqrt{\frac{50 x^{5} y^{3}}{72 x^{4} y}}$ ⓑ $\sqrt[3]{\frac{16 x^{5} y^{7}}{54 x^{2} y^{2}}}$ ⓒ $\sqrt[4]{\frac{5 a^{8} b^{6}}{80 a^{3} b^{2}}} .$

Try It 8.48

Simplify: ⓐ $\sqrt{\frac{48 m^{7} n^{2}}{100 m^{5} n^{8}}}$ ⓑ $\sqrt[3]{\frac{54 x^{7} y^{5}}{250 x^{2} y^{2}}}$ ⓒ $\sqrt[4]{\frac{32 a^{9} b^{7}}{162 a^{3} b^{3}}} .$

In the next example, there is nothing to simplify in the denominators. Since the index on the radicals is the same, we can use the Quotient Property again, to combine them into one radical. We will then look to see if we can simplify the expression.

Example 8.25

Simplify: ⓐ $\frac{\sqrt{48 a^{7}}}{\sqrt{3 a}}$ ⓑ $\frac{\sqrt[3]{−108}}{\sqrt[3]{2}}$ ⓒ $\frac{\sqrt[4]{96 x^{7}}}{\sqrt[4]{3 x^{2}}} .$

Solution

ⓐ

	$\frac{\sqrt{48 a^{7}}}{\sqrt{3 a}}$
The denominator cannot be simplified, so use the Quotient Property to write as one radical.	$\sqrt{\frac{48 a^{7}}{3 a}}$
Simplify the fraction under the radical.	$\sqrt{16 a^{6}}$
Simplify.	$4 \| a^{3} \|$

Step-by-step simplification of a radical expression, demonstrating the process of simplifying quotients of square roots using algebraic properties.

ⓑ

	$\frac{\sqrt[3]{−108}}{\sqrt[3]{2}}$
The denominator cannot be simplified, so use the Quotient Property to write as one radical.	$\sqrt[3]{\frac{−108}{2}}$
Simplify the fraction under the radical.	$\sqrt[3]{−54}$
Rewrite the radicand as a product using perfect cube factors.	$\sqrt[3]{{(−3)}^{3} \cdot 2}$
Rewrite the radical as the product of two radicals.	$\sqrt[3]{{(−3)}^{3}} \cdot \sqrt[3]{2}$
Simplify.	$−3 \sqrt[3]{2}$

Step-by-step simplification of a cube root expression, illustrating the application of radical properties.

ⓒ

	$\frac{\sqrt[4]{96 x^{7}}}{\sqrt[4]{3 x^{2}}}$
The denominator cannot be simplified, so use the Quotient Property to write as one radical.	$\sqrt[4]{\frac{96 x^{7}}{3 x^{2}}}$
Simplify the fraction under the radical.	$\sqrt[4]{32 x^{5}}$
Rewrite the radicand as a product using perfect fourth power factors.	$\sqrt[4]{16 x^{4}} \cdot \sqrt[4]{2 x}$
Rewrite the radical as the product of two radicals.	$\sqrt[4]{{(2 x)}^{4}} \cdot \sqrt[4]{2 x}$
Simplify.	$2 \| x \| \sqrt[4]{2 x}$

Step-by-step simplification of a radical expression by applying the quotient property and factoring out perfect fourth powers.

Try It 8.49

Simplify: ⓐ $\frac{\sqrt{98 z^{5}}}{\sqrt{2 z}}$ ⓑ $\frac{\sqrt[3]{−500}}{\sqrt[3]{2}}$ ⓒ $\frac{\sqrt[4]{486 m^{11}}}{\sqrt[4]{3 m^{5}}} .$

Try It 8.50

Simplify: ⓐ $\frac{\sqrt{128 m^{9}}}{\sqrt{2 m}}$ ⓑ $\frac{\sqrt[3]{−192}}{\sqrt[3]{3}}$ ⓒ $\frac{\sqrt[4]{324 n^{7}}}{\sqrt[4]{2 n^{3}}} .$

Media

Access these online resources for additional instruction and practice with simplifying radical expressions.

Section 8.2 Exercises

Practice Makes Perfect

Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

55.

$\sqrt{27}$

56.

$\sqrt{80}$

57.

$\sqrt{125}$

58.

$\sqrt{96}$

59.

$\sqrt{147}$

60.

$\sqrt{450}$

61.

$\sqrt{800}$

62.

$\sqrt{675}$

63.

ⓐ $\sqrt[4]{32}$ ⓑ $\sqrt[5]{64}$

64.

ⓐ $\sqrt[3]{625}$ ⓑ $\sqrt[6]{128}$

65.

ⓐ $\sqrt[4]{64}$ ⓑ $\sqrt[3]{256}$

66.

ⓐ $\sqrt[4]{3125}$ ⓑ $\sqrt[3]{81}$

In the following exercises, simplify using absolute value signs as needed.

67.

68.

69.

70.

71.

72.

73.

74.

75.

76.

77.

78.

79.

ⓐ $\sqrt[3]{−864}$ ⓑ $\sqrt[4]{−256}$

80.

ⓐ $\sqrt[5]{−486}$ ⓑ $\sqrt[6]{−64}$

81.

ⓐ $\sqrt[5]{−32}$ ⓑ $\sqrt[8]{−1}$

82.

ⓐ $\sqrt[3]{−8}$ ⓑ $\sqrt[4]{−16}$

83.

ⓐ $5 + \sqrt{12}$ ⓑ $\frac{10 - \sqrt{24}}{2}$

84.

ⓐ $8 + \sqrt{96}$ ⓑ $\frac{8 - \sqrt{80}}{4}$

85.

ⓐ $1 + \sqrt{45}$ ⓑ $\frac{3 + \sqrt{90}}{3}$

86.

ⓐ $3 + \sqrt{125}$ ⓑ $\frac{15 + \sqrt{75}}{5}$

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

87.

88.

89.

90.

91.

92.

93.

94.

95.

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

96.

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

97.

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

98.

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

99.

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

100.

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

101.

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

102.

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

103.

104.

105.

106.

107.

ⓐ $\sqrt{\frac{32 x^{5} y^{3}}{18 x^{3} y}}$ ⓑ $\sqrt[3]{\frac{5 x^{6} y^{9}}{40 x^{5} y^{3}}}$ ⓒ $\sqrt[4]{\frac{5 a^{8} b^{6}}{80 a^{3} b^{2}}}$

108.

ⓐ $\sqrt{\frac{75 r^{6} s^{8}}{48 r s^{4}}}$ ⓑ $\sqrt[3]{\frac{24 x^{8} y^{4}}{81 x^{2} y}}$ ⓒ $\sqrt[4]{\frac{32 m^{9} n^{2}}{162 m n^{2}}}$

109.

ⓐ $\sqrt{\frac{27 p^{2} q}{108 p^{4} q^{3}}}$ ⓑ $\sqrt[3]{\frac{16 c^{5} d^{7}}{250 c^{2} d^{2}}}$ ⓒ $\sqrt[6]{\frac{2 m^{9} n^{7}}{128 m^{3} n}}$

110.

ⓐ $\sqrt{\frac{50 r^{5} s^{2}}{128 r^{2} s^{6}}}$ ⓑ $\sqrt[3]{\frac{24 m^{9} n^{7}}{375 m^{4} n^{}}}$ ⓒ $\sqrt[4]{\frac{81 m^{2} n^{8}}{256 m^{1} n^{2}}}$

111.

ⓐ $\frac{\sqrt{45 p^{9}}}{\sqrt{5 q^{2}}}$ ⓑ $\frac{\sqrt[4]{64}}{\sqrt[4]{2}}$ ⓒ $\frac{\sqrt[5]{128 x^{8}}}{\sqrt[5]{2 x^{2}}}$

112.

ⓐ $\frac{\sqrt{80 q^{5}}}{\sqrt{5 q}}$ ⓑ $\frac{\sqrt[3]{−625}}{\sqrt[3]{5}}$ ⓒ $\frac{\sqrt[4]{80 m^{7}}}{\sqrt[4]{5 m}}$

113.

114.

Writing Exercises

115.

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

116.

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

117.

Explain how you know that $\sqrt[5]{x^{10}} = x^{2} .$

118.

Explain why $\sqrt[4]{−64}$ is not a real number but $\sqrt[3]{−64}$ is.

Self Check

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

This table has 3 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “use the product property to simplify radical expressions” and “use the quotient property to simplify radical expressions”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

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

8.2 Simplify Radical Expressions

Learning Objectives

Be Prepared 8.4

Be Prepared 8.5

Be Prepared 8.6

Use the Product Property to Simplify Radical Expressions

Simplified Radical Expression

Product Property of nth Roots

Example 8.13

Simplify Square Roots Using the Product Property of Roots

Solution

Try It 8.25

Try It 8.26

How To

Simplify a radical expression using the Product Property.

Example 8.14

Solution

Try It 8.27

Try It 8.28

Example 8.15

Solution

Try It 8.29

Try It 8.30

Example 8.16

Solution

Try It 8.31

Try It 8.32

Example 8.17

Solution

Try It 8.33

Try It 8.34

Example 8.18

Solution

Try It 8.35

Try It 8.36

Example 8.19

Solution

Try It 8.37

Try It 8.38

Use the Quotient Property to Simplify Radical Expressions

Example 8.20

Solution

Try It 8.39

Try It 8.40

Example 8.21

Solution

Try It 8.41

Try It 8.42

Quotient Property of Radical Expressions

Example 8.22

How to Simplify the Quotient of Radical Expressions

Solution

Try It 8.43

Try It 8.44

How To

Simplify a square root using the Quotient Property.

Example 8.23

Solution

Try It 8.45

Try It 8.46

Example 8.24

Solution

Try It 8.47

Try It 8.48

Example 8.25

Solution

Try It 8.49

Try It 8.50

Media

Section 8.2 Exercises

Practice Makes Perfect

Writing Exercises

Self Check

Product Property of n^th Roots