Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

5.2.3 Radicals and Rational Exponents

5.2.3 • Radicals and Rational Exponents

Activity

If $\sqrt[n]{a^{m}}$ is a real number and $n \geq 2$ , then $\sqrt[n]{a^{m}} = a^{\frac{m}{n}}$ .

So radicals can be rewritten as rational exponents.

With a partner, write each radical as a term with rational exponents on your own paper. Then, turn and talk to your partner.

1. $\sqrt[8]{x}$

Compare your answer:

$x^{\frac{1}{8}}$

2. $\sqrt[3]{x^{2}}$

Compare your answer:

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

3. $\sqrt[11]{x^{5}}$

Compare your answer:

$x^{\frac{5}{11}}$

Now, with a partner, rewrite each term as an equivalent radical.

4. $x^{\frac{1}{2}}$

Compare your answer:

$\sqrt[2]{x}$ or $\sqrt{x}$

5. $x^{\frac{1}{7}}$

Compare your answer:

$\sqrt[7]{x}$

6. $x^{\frac{4}{5}}$

Compare your answer:

$\sqrt[5]{x^{4}}$

Video: Learning About Rational Exponents

Watch the following video to learn more about rational exponents.

Matching Verbal Situations to Expressions and Tables

Self Check

Rewrite

\sqrt[10]{x^7}

as a term with a rational exponent.

$\frac7{10}x$
$10x^7$
$x^\frac{7}{10}$
$x^\frac{10}7$

Additional Resources

Rewriting Radicals

Example 1

When writing radicals as a term with a rational exponent, the exponent becomes the numerator of the fraction in the exponent, the root or index.

In the example above, the 2 is the exponent and becomes the top of the fraction. The 5 is the index and becomes the bottom of the fraction.

Step 1 - Identify the index.

5

Step 2 - Identify the exponent (inside the radical).

2

Step 3 - Write the power over the root.

$8^{\frac{2}{5}}$

Let’s simplify a rational exponent with a variable.

Example 2

$\sqrt[4]{5 x^{3}}$

Step 1 - Identify the index.

4

Step 2 - Identify the exponent (inside the radical).

3

Step 3 - Write the power over the root.

$5 x^{\frac{3}{4}}$

Try it

Rewriting Radicals

Write $\sqrt[3]{12^{2}}$ as a term with a rational exponent.

Compare your answer:

Here is how to rewrite the radical expression:

Step 1 - Identify the index.

3

Step 2 - Identify the exponent (inside the radical).

2

Step 3 - Write the power over the root.

$12^{\frac{2}{3}}$

Write $\sqrt[6]{x^{5}}$ as a term with a rational exponent.

Compare your answer:

Here is how to rewrite the radical expression:

Step 1 - Identify the index.

6

Step 2 - Identify the exponent (inside the radical).

5

Step 3- Write the power over the root.

$x^{\frac{5}{6}}$