Higher Roots - Elementary Algebra

9.7 Higher Roots

Learning Objectives

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

Simplify expressions with higher roots
Use the Product Property to simplify expressions with higher roots
Use the Quotient Property to simplify expressions with higher roots
Add and subtract higher roots

Be Prepared 9.18

Before you get started, take this readiness quiz.

Simplify: $y^{5} y^{4}$ .
If you missed this problem, review Example 6.18.

Be Prepared 9.19

Simplify: ${(n^{2})}^{6}$ .
If you missed this problem, review Example 6.22.

Be Prepared 9.20

Simplify: $\frac{x^{8}}{x^{3}}$ .
If you missed this problem, review Example 6.59.

Simplify Expressions with Higher Roots

Up to now, in this chapter we have worked with squares and square roots. We will now extend our work to include higher powers and higher roots.

Let’s review some vocabulary first.

\begin{matrix} We write: & We say: \\ n^{2} & n squared \\ n^{3} & n cubed \\ n^{4} & n to the fourth \\ n^{5} & n to the fifth \end{matrix}

The terms ‘squared’ and ‘cubed’ come from the formulas for area of a square and volume of a cube.

It will be helpful to have a table of the powers of the integers from $−5 to 5$ . See Figure 9.4.

This figure consists of two tables. The first table shows the results of raising the numbers 1, 2, 3, 4, 5, x, and x squared to the second, third, fourth, and fifth powers. The second table shows the results of raising the numbers negative one through negative five to the second, third, fourth, and fifth powers. The table first has five columns and nine rows. The second has five columns and seven rows. The columns in both tables are labeled, “Number,” “Square,” “Cube,” “Fourth power,” “Fifth power,” nothing, “Number,” “Square,” “Cube,” “Fourth power,” and “Fifth power.” In both tables, the next row reads: n, n squared, n cubed, n to the fourth power, n to the fifth power, nothing, n, n squared, n cubed, n to the fourth power, and n to the fifth power. In the first table, 1 squared, 1 cubed, 1 to the fourth power, and 1 to the fifth power are all shown to be 1. In the next row, 2 squared is 4, 2 cubed is 8, 2 to the fourth power is 16, and 2 to the fifth power is 32. In the next row, 3 squared is 9, 3 cubed is 27, 3 to the fourth power is 81, and 3 to the fifth power is 243. In the next row, 4 squared is 16, 4 cubed is 64, 4 to the fourth power is 246, and 4 to the fifth power is 1024. In the next row, 5 squared is 25, 5 cubed is 125, 5 to the fourth power is 625, and 5 to the fifth power is 3125. In the next row, x squared, x cubed, x to the fourth power, and x to the fifth power are listed. In the next row, x squared squared is x to the fourth power, x cubed squared is x to the fifth power, x squared to the fourth power is x to the eighth power, and x squared to the fifth power is x to the tenth power. In the second table, negative 1 squared is 1, negative 1 cubed is negative 1, negative 1 to the fourth power is 1, and negative 1 to the fifth power is negative 1. In the next row, negative 2 squared is 4, negative 2 cubed is negative 8, negative 2 to the fourth power is 16, and negative 2 to the fifth power is negative 32. In the next row, negative 4 squared is 16, negative 4 cubed is negative 64, negative 4 to the fourth power is 256, and negative 4 to the fifth power is negative 1024. In the next row, negative 5 squared is 25, negative 5 cubed is negative 125, negative 5 to the fourth power is 625, and negative 5 to the fifth power is negative 3125.

Figure 9.4 First through fifth powers of integers from

−5

5 .

Notice the signs in Figure 9.4. All powers of positive numbers are positive, of course. But when we have a negative number, the even powers are positive and the odd powers are negative. We’ll copy the row with the powers of $−2$ below to help you see this.

Earlier in this chapter we defined the square root of a number.

If n^{2} = m, then n is a square root of m .

And we have used the notation $\sqrt{m}$ to denote the principal square root. So $\sqrt{m} \geq 0$ always.

We will now extend the definition to higher roots.

nth Root of a Number

If $b^{n} = a$ , then $b$ is an nth root of a number $a$ .

The principal nth root of $a$ is written $\sqrt[n]{a}$ .

n is called the index of the radical.

We do not write the index for a square root. Just like we use the word ‘cubed’ for $b^{3}$ , we use the term ‘cube root’ for $\sqrt[3]{a}$ .

We refer to Figure 9.4 to help us find higher roots.

\begin{matrix} 4^{3} & = & 64 & \sqrt[3]{64} & = & 4 \\ 3^{4} & = & 81 & \sqrt[4]{81} & = & 3 \\ {(−2)}^{5} & = & −32 & \sqrt[5]{−32} & = & −2 \end{matrix}

Could we have an even root of a negative number? No. We know that the square root of a negative number is not a real number. The same is true for any even root. Even roots of negative numbers are not real numbers. Odd roots of negative numbers are real numbers.

Properties of $\sqrt[n]{a}$

When $n$ is an even number and

$a \geq 0$ , then $\sqrt[n]{a}$ is a real number
$a < 0$ , then $\sqrt[n]{a}$ is not a real number

When $n$ is an odd number, $\sqrt[n]{a}$ is a real number for all values of $a$ .

Example 9.88

Simplify: ⓐ $\sqrt[3]{8}$ ⓑ $\sqrt[4]{81}$ ⓒ $\sqrt[5]{32}$ .

Solution

ⓐ

	$\sqrt[3]{8}$
Since ${(2)}^{3} = 8 .$	$2$

This table illustrates finding the cube root of 8, demonstrating its value and the mathematical reason.

ⓑ

	$\sqrt[4]{81}$
Since ${(3)}^{4} = 81 .$	$3$

This table presents the fourth root of 81, its resulting value, and the mathematical explanation for the solution.

ⓒ

	$\sqrt[5]{32}$
Since ${(2)}^{5} = 32 .$	$2$

Example showing the evaluation of the fifth root of 32, with its mathematical justification.

Try It 9.175

Simplify: ⓐ $\sqrt[3]{27}$ ⓑ $\sqrt[4]{256}$ ⓒ $\sqrt[5]{243}$ .

Try It 9.176

Simplify: ⓐ $\sqrt[3]{1000}$ ⓑ $\sqrt[4]{16}$ ⓒ $\sqrt[5]{32}$ .

Example 9.89

Simplify: ⓐ $\sqrt[3]{−64}$ ⓑ $\sqrt[4]{−16}$ ⓒ $\sqrt[5]{−243}$ .

Solution

ⓐ

	$\sqrt[3]{−64}$
Since ${(−4)}^{3} = −64 .$	$−4$

Demonstrates the calculation of the cube root of -64 with a step-by-step explanation.

ⓑ

	$\sqrt[4]{−16}$
Think, ${(?)}^{4} = −16 .$ No real number raised to the fourth power is negative.	Not a real number.

This table demonstrates why the fourth root of -16 is not a real number, providing the mathematical expression, the reasoning, and the conclusion.

ⓒ

	$\sqrt[5]{−243}$
Since ${(−3)}^{5} = −243 .$	$−3$

Calculation of the fifth root of -243, including the expression, explanation, and final solution.

Try It 9.177

Simplify: ⓐ $\sqrt[3]{−125}$ ⓑ $\sqrt[4]{−16}$ ⓒ $\sqrt[5]{−32}$ .

Try It 9.178

Simplify: ⓐ $\sqrt[3]{−216}$ ⓑ $\sqrt[4]{−81}$ ⓒ $\sqrt[5]{−1024}$ .

When we worked with square roots that had variables in the radicand, we restricted the variables to non-negative values. Now we will remove this restriction.

The odd root of a number can be either positive or negative. We have seen that $\sqrt[3]{−64} = −4$ .

But the even root of a non-negative number is always non-negative, because we take the principal nth root.

Suppose we start with $a = −5$ .

\begin{matrix} {(−5)}^{4} = 625 & \sqrt[4]{625} = 5 \end{matrix}

How can we make sure the fourth root of −5 raised to the fourth power, ${(−5)}^{4}$ is 5? We will see in the following property.

Simplifying Odd and Even Roots

For any integer $n \geq 2$ ,

\begin{matrix} when n is odd & \sqrt[n]{a^{n}} = a \\ when n is even & \sqrt[n]{a^{n}} = | a | \end{matrix}

We must use the absolute value signs when we take an even root of an expression with a variable in the radical.

Example 9.90

Simplify: ⓐ $\sqrt{x^{2}}$ ⓑ $\sqrt[3]{n^{3}}$ ⓒ $\sqrt[4]{p^{4}}$ ⓓ $\sqrt[5]{y^{5}}$ .

Solution

We use the absolute value to be sure to get the positive root.

ⓐ

	$\sqrt{x^{2}}$
Since ${(x)}^{2} = x^{2}$ and we want the positive root.	$\| x \|$

Illustration of simplifying sqrt(x^2) to |x| and the underlying mathematical justification.

ⓑ

	$\sqrt[3]{n^{3}}$
Since ${(n)}^{3} = n^{3} .$ It is an odd root so there is no need for an absolute value sign.	$n$

This table illustrates the simplification of the cube root of n cubed, providing the mathematical reasoning for the derivation.

ⓒ

	$\sqrt[4]{p^{4}}$
Since ${(p)}^{4} = p^{4}$ and we want the positive root.	$\| p \|$

Illustrates the simplification of the fourth root of p to the fourth power to the absolute value of p, with an explanation.

ⓓ

	$\sqrt[5]{y^{5}}$
Since ${(y)}^{5} = y^{5} .$ It is an odd root so there is no need for an absolute value sign.	$y$

This table demonstrates the simplification of the fifth root of y^5 to y, explaining why an absolute value is not required for odd roots.

Try It 9.179

Simplify: ⓐ $\sqrt{b^{2}}$ ⓑ $\sqrt[3]{w^{3}}$ ⓒ $\sqrt[4]{m^{4}}$ ⓓ $\sqrt[5]{q^{5}}$ .

Try It 9.180

Simplify: ⓐ $\sqrt{y^{2}}$ ⓑ $\sqrt[3]{p^{3}}$ ⓒ $\sqrt[4]{z^{4}}$ ⓓ $\sqrt[5]{q^{5}}$ .

Example 9.91

Simplify: ⓐ $\sqrt[3]{y^{18}}$ ⓑ $\sqrt[4]{z^{8}}$ .

Solution

ⓐ

	$\sqrt[3]{y^{18}}$
Since ${(y^{6})}^{3} = y^{18} .$	$\sqrt[3]{{(y^{6})}^{3}}$
	$y^{6}$

This table demonstrates the step-by-step algebraic simplification of the cube root of y to the power of 18.

ⓑ

	$\sqrt[4]{z^{8}}$
Since ${(z^{2})}^{4} = z^{8} .$	$\sqrt[4]{{(z^{2})}^{4}}$
Since $z^{2}$ is positive, we do not need an absolute value sign.	$z^{2}$

Step-by-step simplification of the radical expression 4th root of z to the power of 8, showing mathematical reasons.

Try It 9.181

Simplify: ⓐ $\sqrt[4]{u^{12}}$ ⓑ $\sqrt[3]{v^{15}}$ .

Try It 9.182

Simplify: ⓐ $\sqrt[5]{c^{20}}$ ⓑ $\sqrt[6]{d^{24}}$ .

Example 9.92

Simplify: ⓐ $\sqrt[3]{64 p^{6}}$ ⓑ $\sqrt[4]{16 q^{12}}$ .

Solution

ⓐ

	$\sqrt[3]{64 p^{6}}$
Rewrite $64 p^{6} as {(4 p^{2})}^{3} .$	$\sqrt[3]{{(4 p^{2})}^{3}}$
Take the cube root.	$4 p^{2}$

Illustrates the step-by-step process of simplifying a cube root expression with a perfect cube radicand.

ⓑ

	$\sqrt[4]{16 q^{12}}$
Rewrite the radicand as a fourth power.	$\sqrt[4]{{(2 q^{3})}^{4}}$
Take the fourth root.	$2 \| q^{3} \|$

Steps to simplify the fourth root of 16q^12, showing intermediate algebraic expressions for each operation.

Try It 9.183

Simplify: ⓐ $\sqrt[3]{27 x^{27}}$ ⓑ $\sqrt[4]{81 q^{28}}$ .

Try It 9.184

Simplify: ⓐ $\sqrt[3]{125 p^{9}}$ ⓑ $\sqrt[5]{243 q^{25}}$ .

Use the Product Property to Simplify Expressions with Higher Roots

We will simplify expressions with higher roots in much the same way as we simplified expressions with square roots. An nth root is considered simplified if it has no factors of $m^{n}$ .

Simplified nth Root

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

We will generalize the Product Property of Square Roots to include any integer root $n \geq 2$ .

Product Property of nth Roots

\begin{matrix} \sqrt[n]{a b} = \sqrt[n]{a} \cdot \sqrt[n]{b} & and & \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a b} \end{matrix}

when $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers and for any integer $n \geq 2$

Example 9.93

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

Solution

ⓐ

	$\sqrt[3]{x^{4}}$
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.	$x \sqrt[3]{x}$

Steps for simplifying the cube root of x^4 by factoring the radicand and extracting perfect cubes.

ⓑ

	$\sqrt[4]{x^{7}}$
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.	$\| x \| \sqrt[4]{x^{3}}$

Illustrates the step-by-step simplification of the radical expression fourth root of x to the power of seven.

Try It 9.185

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

Try It 9.186

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

Example 9.94

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

Solution

ⓐ

	$\sqrt[3]{16}$
	$\sqrt[3]{2^{4}}$
Rewrite the radicand as a product using the greatest perfect cube factor.	$\sqrt[3]{2^{3} \cdot 2}$
Rewrite the radical as the product of two radicals.	$\sqrt[3]{2^{3}} \cdot \sqrt[3]{2}$
Simplify.	$2 \sqrt[3]{2}$

Step-by-step process demonstrating the simplification of the cube root of 16.

ⓑ

	$\sqrt[4]{243}$
	$\sqrt[4]{3^{5}}$
Rewrite the radicand as a product using the greatest perfect fourth power factor.	$\sqrt[4]{3^{4} \cdot 3}$
Rewrite the radical as the product of two radicals.	$\sqrt[4]{3^{4}} \cdot \sqrt[4]{3}$
Simplify.	$3 \sqrt[4]{3}$

Step-by-step simplification of the fourth root of 243.

Try It 9.187

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

Try It 9.188

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

Don’t forget to use the absolute value signs when taking an even root of an expression with a variable in the radical.

Example 9.95

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

Solution

ⓐ

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

This table details the step-by-step simplification of the cube root expression (24x^7)^(1/3) to its simplified form 2x^2 * (3x)^(1/3).

ⓑ

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

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

Try It 9.189

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

Try It 9.190

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

Example 9.96

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.	$−3$

Steps to calculate the cube root of -27.

ⓑ

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

This table demonstrates why the fourth root of -16 is not considered a real number.

Try It 9.191

Simplify: ⓐ $\sqrt[3]{−108}$ ⓑ $\sqrt[4]{−48}$ .

Try It 9.192

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

Use the Quotient Property to Simplify Expressions with Higher Roots

We can simplify higher roots with quotients in the same way we simplified square roots. First we simplify any fractions inside the radical.

Example 9.97

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

Solution

ⓐ

	$\sqrt[3]{\frac{a^{8}}{a^{5}}}$
Simplify the fraction under the radical first.	$\sqrt[3]{a^{3}}$
Simplify.	$a$

Step-by-step simplification of the radical expression ³√(a⁸/a⁵) to 'a', illustrating algebraic simplification.

ⓑ

	$\sqrt[4]{\frac{a^{10}}{a^{2}}}$
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 process demonstrating the simplification of a radical expression involving variables.

Try It 9.193

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

Try It 9.194

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

Previously, we used the Quotient Property ‘in reverse’ to simplify square roots. Now we will generalize the formula to include higher roots.

Quotient Property of nth Roots

\begin{matrix} \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}} \end{matrix}

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

Example 9.98

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

Solution

ⓐ

	$\frac{\sqrt[3]{−108}}{\sqrt[3]{2}}$
Neither radicand is a perfect cube, 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 using radical properties.

ⓑ

	$\frac{\sqrt[4]{96 x^{7}}}{\sqrt[4]{3 x^{2}}}$
Neither radicand is a perfect fourth power, 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.	$\sqrt[4]{2^{4} x^{4} \cdot 2 x}$
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 involving the division of fourth roots.

Try It 9.195

Simplify: ⓐ $\frac{\sqrt[3]{−532}}{\sqrt[3]{2}}$ ⓑ $\frac{\sqrt[4]{486 m^{11}}}{\sqrt[4]{3 m^{5}}}$ .

Try It 9.196

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

If the fraction inside the radical cannot be simplified, we use the first form of the Quotient Property to rewrite the expression as the quotient of two radicals.

Example 9.99

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

Solution

ⓐ

	$\sqrt[3]{\frac{24 x^{7}}{y^{3}}}$
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.	$\frac{\sqrt[3]{{(2 x^{2})}^{3}} \sqrt[3]{3 x}}{\sqrt[3]{y^{3}}}$
Simplify.	$\frac{2 x^{2} \sqrt[3]{3 x}}{y}$

Step-by-step process for simplifying a cube root expression involving a fraction and variables.

ⓑ

	$\sqrt[4]{\frac{48 x^{10}}{y^{8}}}$
The fraction in the radicand cannot be simplified. Use the Quotient Property to write as two radicals.	$\frac{\sqrt[4]{48 x^{10}}}{\sqrt[4]{y^{8}}}$
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.	$\frac{\sqrt[4]{{(2 x^{2})}^{4}} \sqrt[4]{3 x^{2}}}{\sqrt[4]{{(y^{2})}^{4}}}$
Simplify.	$\frac{2 x^{2} \sqrt[4]{3 x^{2}}}{y^{2}}$

Step-by-step process demonstrating the simplification of a fourth root algebraic expression using radical properties.

Try It 9.197

Simplify: ⓐ $\sqrt[3]{\frac{108 c^{10}}{d^{6}}}$ ⓑ $\sqrt[4]{\frac{80 x^{10}}{y^{5}}}$ .

Try It 9.198

Simplify: ⓐ $\sqrt[3]{\frac{40 r^{3}}{s}}$ ⓑ $\sqrt[4]{\frac{162 m^{14}}{n^{12}}}$ .

Add and Subtract Higher Roots

We can add and subtract higher roots like we added and subtracted square roots. First we provide a formal definition of like radicals.

Like Radicals

Radicals with the same index and same radicand are called like radicals.

Like radicals have the same index and the same radicand.

$9 \sqrt[4]{42 x}$ and $−2 \sqrt[4]{42 x}$ are like radicals.
$5 \sqrt[3]{125 x}$ and $6 \sqrt[3]{125 y}$ are not like radicals. The radicands are different.
$2 \sqrt[5]{1000 q}$ and $−4 \sqrt[4]{1000 q}$ are not like radicals. The indices are different.

We add and subtract like radicals in the same way we add and subtract like terms. We can add $9 \sqrt[4]{42 x} + (−2 \sqrt[4]{42 x})$ and the result is $7 \sqrt[4]{42 x}$ .

Example 9.100

Simplify: ⓐ $\sqrt[3]{4 x} + \sqrt[3]{4 x}$ ⓑ $4 \sqrt[4]{8} - 2 \sqrt[4]{8}$ .

Solution

ⓐ

	$\sqrt[3]{4 x} + \sqrt[3]{4 x}$
The radicals are like, so we add the coefficients.	$2 \sqrt[3]{4 x}$

Example illustrating the addition of like radicals, including an explanation and the simplified expression.

ⓑ

	$4 \sqrt[4]{8} - 2 \sqrt[4]{8}$
The radicals are like, so we subtract the coefficients.	$2 \sqrt[4]{8}$

Simplification of like radical expressions, demonstrating the subtraction of coefficients with an accompanying explanation.

Try It 9.199

Simplify: ⓐ $\sqrt[5]{3 x} + \sqrt[5]{3 x}$ ⓑ $3 \sqrt[3]{9} - \sqrt[3]{9}$ .

Try It 9.200

Simplify: ⓐ $\sqrt[4]{10 y} + \sqrt[4]{10 y}$ ⓑ $5 \sqrt[6]{32} - 3 \sqrt[6]{32}$ .

When an expression does not appear to have like radicals, we will simplify each radical first. Sometimes this leads to an expression with like radicals.

Example 9.101

Simplify: ⓐ $\sqrt[3]{54} - \sqrt[3]{16}$ ⓑ $\sqrt[4]{48} + \sqrt[4]{243}$ .

Solution

ⓐ

	$\sqrt[3]{54} - \sqrt[3]{16}$
Rewrite each radicand using perfect cube factors.	$\sqrt[3]{27} \cdot \sqrt[3]{2} - \sqrt[3]{8} \cdot \sqrt[3]{2}$
Rewrite the perfect cubes.	$\sqrt[3]{{(3)}^{3}} \sqrt[3]{2} - \sqrt[3]{{(2)}^{3}} \sqrt[3]{2}$
Simplify the radicals where possible.	$3 \sqrt[3]{2} - 2 \sqrt[3]{2}$
Combine like radicals.	$\sqrt[3]{2}$

Step-by-step simplification of an expression involving cube roots.

ⓑ

	$\sqrt[4]{48} + \sqrt[4]{243}$
Rewrite using perfect fourth power factors.	$\sqrt[4]{16} \cdot \sqrt[4]{3} + \sqrt[4]{81} \cdot \sqrt[4]{3}$
Rewrite the perfect fourth powers.	$\sqrt[4]{{(2)}^{4}} \sqrt[4]{3} + \sqrt[4]{{(3)}^{4}} \sqrt[4]{3}$
Simplify the radicals where possible.	$2 \sqrt[4]{3} + 3 \sqrt[4]{3}$
Combine like radicals.	$5 \sqrt[4]{3}$

Step-by-step simplification of a sum of fourth roots by factoring out perfect fourth powers and combining like radicals.

Try It 9.201

Simplify: ⓐ $\sqrt[3]{192} - \sqrt[3]{81}$ ⓑ $\sqrt[4]{32} + \sqrt[4]{512}$ .

Try It 9.202

Simplify: ⓐ $\sqrt[3]{128} - \sqrt[3]{250}$ ⓑ $\sqrt[5]{64} + \sqrt[5]{486}$ .

Example 9.102

Simplify: ⓐ $\sqrt[3]{24 x^{4}} - \sqrt[3]{−81 x^{7}}$ ⓑ $\sqrt[4]{162 y^{9}} + \sqrt[4]{512 y^{5}}$ .

Solution

ⓐ

	$\sqrt[3]{24 x^{4}} - \sqrt[3]{−81 x^{7}}$
Rewrite each radicand using perfect cube factors.	$\sqrt[3]{8 x^{3}} \cdot \sqrt[3]{3 x} - \sqrt[3]{−27 x^{6}} \cdot \sqrt[3]{3 x}$
Rewrite the perfect cubes.	$\sqrt[3]{{(2 x)}^{3}} \sqrt[3]{3 x} - \sqrt[3]{{(−3 x^{2})}^{3}} \sqrt[3]{3 x}$
Simplify the radicals where possible.	$2 x \sqrt[3]{3 x} - (−3 x^{2} \sqrt[3]{3 x})$

Step-by-step simplification of an algebraic expression involving cube roots.

ⓑ

	$\sqrt[4]{162 y^{9}} + \sqrt[4]{512 y^{5}}$
Rewrite each radicand using perfect fourth power factors.	$\sqrt[4]{81 y^{8}} \cdot \sqrt[4]{2 y} + \sqrt[4]{256 y^{4}} \cdot \sqrt[4]{2 y}$
Rewrite the perfect fourth powers.	$\sqrt[4]{{(3 y^{2})}^{4}} \cdot \sqrt[4]{2 y} + \sqrt[4]{{(4 y)}^{4}} \cdot \sqrt[4]{2 y}$
Simplify the radicals where possible.	$3 y^{2} \sqrt[4]{2 y} + 4 \| y \| \sqrt[4]{2 y}$

Step-by-step simplification of a sum involving fourth-root radical expressions.

Try It 9.203

Simplify: ⓐ $\sqrt[3]{32 y^{5}} - \sqrt[3]{−108 y^{8}}$ ⓑ $\sqrt[4]{243 r^{11}} + \sqrt[4]{768 r^{10}}$ .

Try It 9.204

Simplify: ⓐ $\sqrt[3]{40 z^{7}} - \sqrt[3]{−135 z^{4}}$ ⓑ $\sqrt[4]{80 s^{13}} + \sqrt[4]{1280 s^{6}}$ .

Media

Access these online resources for additional instruction and practice with simplifying higher roots.

Section 9.7 Exercises

Practice Makes Perfect

Simplify Expressions with Higher Roots

In the following exercises, simplify.

442.

ⓐ $\sqrt[3]{216}$ ⓑ $\sqrt[4]{256}$ ⓒ $\sqrt[5]{32}$

443.

ⓐ $\sqrt[3]{27}$ ⓑ $\sqrt[4]{16}$ ⓒ $\sqrt[5]{243}$

444.

ⓐ $\sqrt[3]{512}$ ⓑ $\sqrt[4]{81}$ ⓒ $\sqrt[5]{1}$

445.

ⓐ $\sqrt[3]{125}$ ⓑ $\sqrt[4]{1296}$ ⓒ $\sqrt[5]{1024}$

446.

ⓐ $\sqrt[3]{−8}$ ⓑ $\sqrt[4]{−81}$ ⓒ $\sqrt[5]{−32}$

447.

ⓐ $\sqrt[3]{−64}$ ⓑ $\sqrt[4]{−16}$ ⓒ $\sqrt[5]{−243}$

448.

ⓐ $\sqrt[3]{−125}$ ⓑ $\sqrt[4]{−1296}$ ⓒ $\sqrt[5]{−1024}$

449.

ⓐ $\sqrt[3]{−512}$ ⓑ $\sqrt[4]{−81}$ ⓒ $\sqrt[5]{−1}$

450.

ⓐ $\sqrt[5]{u^{5}}$ ⓑ $\sqrt[8]{v^{8}}$

451.

ⓐ $\sqrt[3]{a^{3}}$
ⓑ

452.

ⓐ $\sqrt[4]{y^{4}}$ ⓑ $\sqrt[7]{m^{7}}$

453.

ⓐ $\sqrt[8]{k^{8}}$ ⓑ $\sqrt[6]{p^{6}}$

454.

ⓐ $\sqrt[3]{x^{9}}$ ⓑ $\sqrt[4]{y^{12}}$

455.

ⓐ $\sqrt[5]{a^{10}}$ ⓑ $\sqrt[3]{b^{27}}$

456.

ⓐ $\sqrt[4]{m^{8}}$ ⓑ $\sqrt[5]{n^{20}}$

457.

ⓐ $\sqrt[6]{r^{12}}$ ⓑ $\sqrt[3]{s^{30}}$

458.

ⓐ $\sqrt[4]{16 x^{8}}$ ⓑ $\sqrt[6]{64 y^{12}}$

459.

ⓐ $\sqrt[3]{−8 c^{9}}$ ⓑ $\sqrt[3]{125 d^{15}}$

460.

ⓐ $\sqrt[3]{216 a^{6}}$ ⓑ $\sqrt[5]{32 b^{20}}$

461.

ⓐ $\sqrt[7]{128 r^{14}}$ ⓑ $\sqrt[4]{81 s^{24}}$

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

462.

ⓐ $\sqrt[3]{r^{5}}$ ⓑ $\sqrt[4]{s^{10}}$

463.

ⓐ $\sqrt[5]{u^{7}}$ ⓑ $\sqrt[6]{v^{11}}$

464.

ⓐ $\sqrt[4]{m^{5}}$ ⓑ $\sqrt[8]{n^{10}}$

465.

ⓐ $\sqrt[5]{p^{8}}$ ⓑ $\sqrt[3]{q^{8}}$

466.

ⓐ $\sqrt[4]{32}$ ⓑ $\sqrt[7]{64}$

467.

ⓐ $\sqrt[3]{625}$ ⓑ $\sqrt[6]{128}$

468.

ⓐ $\sqrt[5]{64}$ ⓑ $\sqrt[3]{256}$

469.

ⓐ $\sqrt[4]{3125}$ ⓑ $\sqrt[3]{81}$

470.

ⓐ $\sqrt[3]{108 x^{5}}$ ⓑ $\sqrt[4]{48 y^{6}}$

471.

ⓐ $\sqrt[5]{96 a^{7}}$ ⓑ $\sqrt[3]{375 b^{4}}$

472.

ⓐ $\sqrt[4]{405 m^{10}}$ ⓑ $\sqrt[5]{160 n^{8}}$

473.

ⓐ $\sqrt[3]{512 p^{5}}$ ⓑ $\sqrt[4]{324 q^{7}}$

474.

ⓐ $\sqrt[3]{−864}$ ⓑ $\sqrt[4]{−256}$

475.

ⓐ $\sqrt[5]{−486}$ ⓑ $\sqrt[6]{−64}$

476.

ⓐ $\sqrt[5]{−32}$ ⓑ $\sqrt[8]{−1}$

477.

ⓐ $\sqrt[3]{−8}$ ⓑ $\sqrt[4]{−16}$

Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

478.

ⓐ $\sqrt[3]{\frac{p^{11}}{p^{2}}}$ ⓑ $\sqrt[4]{\frac{q^{17}}{q^{13}}}$

479.

ⓐ $\sqrt[5]{\frac{d^{12}}{d^{7}}}$ ⓑ $\sqrt[8]{\frac{m^{12}}{m^{4}}}$

480.

ⓐ $\sqrt[5]{\frac{u^{21}}{u^{11}}}$ ⓑ $\sqrt[6]{\frac{v^{30}}{v^{12}}}$

481.

ⓐ $\sqrt[3]{\frac{r^{14}}{r^{5}}}$ ⓑ $\sqrt[4]{\frac{c^{21}}{c^{9}}}$

482.

ⓐ $\frac{\sqrt[4]{64}}{\sqrt[4]{2}}$ ⓑ $\frac{\sqrt[5]{128 x^{8}}}{\sqrt[5]{2 x^{2}}}$

483.

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

484.

ⓐ $\sqrt[3]{\frac{1250}{2}}$ ⓑ $\sqrt[4]{\frac{486 y^{9}}{2 y^{3}}}$

485.

ⓐ $\sqrt[3]{\frac{162}{6}}$ ⓑ $\sqrt[4]{\frac{160 r^{10}}{5 r^{3}}}$

486.

ⓐ $\sqrt[3]{\frac{54 a^{8}}{b^{3}}}$ ⓑ $\sqrt[4]{\frac{64 c^{5}}{d^{2}}}$

487.

ⓐ $\sqrt[5]{\frac{96 r^{11}}{s^{3}}}$ ⓑ $\sqrt[6]{\frac{128 u^{7}}{v^{3}}}$

488.

ⓐ $\sqrt[3]{\frac{81 s^{8}}{t^{3}}}$ ⓑ $\sqrt[4]{\frac{64 p^{15}}{q^{12}}}$

489.

ⓐ $\sqrt[3]{\frac{625 u^{10}}{v^{3}}}$ ⓑ $\sqrt[4]{\frac{729 c^{21}}{d^{8}}}$

Add and Subtract Higher Roots

In the following exercises, simplify.

490.

ⓐ $\sqrt[7]{8 p} + \sqrt[7]{8 p}$ ⓑ $3 \sqrt[3]{25} - \sqrt[3]{25}$

491.

ⓐ $\sqrt[3]{15 q} + \sqrt[3]{15 q}$ ⓑ $2 \sqrt[4]{27} - 6 \sqrt[4]{27}$

492.

ⓐ $3 \sqrt[5]{9 x} + 7 \sqrt[5]{9 x}$ ⓑ $8 \sqrt[7]{3 q} - 2 \sqrt[7]{3 q}$

493.

ⓐ

ⓑ

494.

ⓐ $\sqrt[3]{81} - \sqrt[3]{192}$ ⓑ $\sqrt[4]{512} - \sqrt[4]{32}$

495.

ⓐ $\sqrt[3]{250} - \sqrt[3]{54}$ ⓑ $\sqrt[4]{243} - \sqrt[4]{1875}$

496.

ⓐ $\sqrt[3]{128} + \sqrt[3]{250}$ ⓑ $\sqrt[5]{729} + \sqrt[5]{96}$

497.

ⓐ $\sqrt[4]{243} + \sqrt[4]{1250}$ ⓑ $\sqrt[3]{2000} + \sqrt[3]{54}$

498.

ⓐ $\sqrt[3]{64 a^{10}} - \sqrt[3]{−216 a^{12}}$ ⓑ $\sqrt[4]{486 u^{7}} + \sqrt[4]{768 u^{3}}$

499.

ⓐ $\sqrt[3]{80 b^{5}} - \sqrt[3]{−270 b^{3}}$ ⓑ $\sqrt[4]{160 v^{10}} - \sqrt[4]{1280 v^{3}}$

Mixed Practice

In the following exercises, simplify.

500.

$\sqrt[4]{16}$

501.

$\sqrt[6]{64}$

502.

$\sqrt[3]{a^{3}}$

503.

504.

$\sqrt[3]{−8 c^{9}}$

505.

$\sqrt[3]{125 d^{15}}$

506.

$\sqrt[3]{r^{5}}$

507.

$\sqrt[4]{s^{10}}$

508.

$\sqrt[3]{108 x^{5}}$

509.

$\sqrt[4]{48 y^{6}}$

510.

$\sqrt[5]{−486}$

511.

$\sqrt[6]{−64}$

512.

$\frac{\sqrt[4]{64}}{\sqrt[4]{2}}$

513.

$\frac{\sqrt[5]{128 x^{8}}}{\sqrt[5]{2 x^{2}}}$

514.

$\sqrt[5]{\frac{96 r^{11}}{s^{3}}}$

515.

$\sqrt[6]{\frac{128 u^{7}}{v^{3}}}$

516.

$\sqrt[3]{81} - \sqrt[3]{192}$

517.

$\sqrt[4]{512} - \sqrt[4]{32}$

518.

$\sqrt[3]{64 a^{10}} - \sqrt[3]{−216 a^{12}}$

519.

$\sqrt[4]{486 u^{7}} + \sqrt[4]{768 u^{3}}$

Everyday Math

520.

Population growth The expression $10 \cdot x^{n}$ models the growth of a mold population after $n$ generations. There were 10 spores at the start, and each had $x$ offspring. So $10 \cdot x^{5}$ is the number of offspring at the fifth generation. At the fifth generation there were 10,240 offspring. Simplify the expression $\sqrt[5]{\frac{10,240}{10}}$ to determine the number of offspring of each spore.

521.

Spread of a virus The expression $3 \cdot x^{n}$ models the spread of a virus after $n$ cycles. There were three people originally infected with the virus, and each of them infected $x$ people. So $3 \cdot x^{4}$ is the number of people infected on the fourth cycle. At the fourth cycle 1875 people were infected. Simplify the expression $\sqrt[4]{\frac{1875}{3}}$ to determine the number of people each person infected.

Writing Exercises

522.

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

523.

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 four columns and five rows. The first row labels each column: “I can…,” “Confidentaly,” “With some help,” and “No – I don’t get it!” The rows under the “I can…,” column read, “simplify expressions with hither roots.,” “use the product property to simplify expressions with higher roots.,” “use the quotient property to simplify expressions with higher roots.,” and “add and subtract higher roots.” The rest of the rows under the columns are empty.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

9.7 Higher Roots

Learning Objectives

Be Prepared 9.18

Be Prepared 9.19

Be Prepared 9.20

Simplify Expressions with Higher Roots

nth Root of a Number

Properties of a n a n

Example 9.88

Solution

Try It 9.175

Try It 9.176

Example 9.89

Solution

Try It 9.177

Try It 9.178

Simplifying Odd and Even Roots

Example 9.90

Solution

Try It 9.179

Try It 9.180

Example 9.91

Solution

Try It 9.181

Try It 9.182

Example 9.92

Solution

Try It 9.183

Try It 9.184

Use the Product Property to Simplify Expressions with Higher Roots

Simplified nth Root

Product Property of nth Roots

Example 9.93

Solution

Try It 9.185

Try It 9.186

Example 9.94

Solution

Try It 9.187

Try It 9.188

Example 9.95

Solution

Try It 9.189

Try It 9.190

Example 9.96

Solution

Try It 9.191

Try It 9.192

Use the Quotient Property to Simplify Expressions with Higher Roots

Example 9.97

Solution

Try It 9.193

Try It 9.194

Quotient Property of nth Roots

Example 9.98

Solution

Try It 9.195

Try It 9.196

Example 9.99

Solution

Try It 9.197

Try It 9.198

Add and Subtract Higher Roots

Like Radicals

Example 9.100

Solution

Try It 9.199

Try It 9.200

Example 9.101

Solution

Try It 9.201

Try It 9.202

Example 9.102

Solution

Try It 9.203

Try It 9.204

Media

Section 9.7 Exercises

Practice Makes Perfect

Everyday Math

Properties of $\sqrt[n]{a}$