How to Solve Rational Exponents and Radicals?

An expression with a rational exponent is equivalent to a radical in which the denominator is the index and the numerator is the exponent. Any radical expression can be written with a rational exponent, which we call exponential form.

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• How to Solve Rational Exponents

A step-by-step guide to rational exponents and radicals

By identifying powers and roots and converting them to radicals, we can write rational power exponents expressions as radicals. Consider the rational exponents’ expression \(a^{\frac{m}{n}}\). Now, follow the steps given:

1. Identify power by looking at the numerator of the rational exponent. Here in the rational exponent \(a^{\frac{m}{n}}\), \(m\) is the power.
2. Identify the root by looking at the denominator of the rational exponent. Here in rational exponent \(a^{\frac{m}{n}}\), \(n\) is the root.
3. Write the base as the radicand, power raising to the radicand, and the root as the index. Here we can write \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\).

We can also convert radicals into rational exponents. Consider the square root of the positive number \(\sqrt{a}\). We can write the square of \(\sqrt{a}\) as a rational exponent. \(\sqrt{a}=a^{\frac{1}{2}}\) which is a rational exponent.

Rational Exponents and Radicals, Example 1:

Express \((2x)^{\frac{1}{3}}\) in radical form.

Solution:

\((2x)^{\frac{1}{3}}\) \(=\)\(\sqrt[3]{2x}\)

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Rational Exponents and Radicals, Example 2:

Write \(\sqrt[4]{81}\) as an expression with a rational exponent.

Solution:

\(\sqrt[4]{81}=81^{\frac{1}{4}}\)

Exercises for Rational Exponents and Radicals

Rewrite the expression in radical form or rational exponent.

1. \(\color{blue}{6\sqrt[3]{xy}}\)
2. \(\color{blue}{27^{\frac{2}{3}}}\)
3. \(\color{blue}{\sqrt[2]{\left(5ab\right)^3}}\)
4. \(\color{blue}{\left(10\:r\right)^{-\frac{3}{4}}}\)
5. \(\color{blue}{\frac{1}{\sqrt[5]{\left(6x\right)^3}}}\)

1. \(\color{blue}{6\left(xy\right)^{\frac{1}{3}}}\)
2. \(\color{blue}{\sqrt[3]{27^2}}\)
3. \(\color{blue}{\left(5ab\right)^{\frac{3}{2}}}\)
4. \(\color{blue}{\frac{1}{\sqrt[4]{\left(10\:r\right)^3}}}\)
5. \(\color{blue}{\left(6x\right)^{-\frac{3}{5}}}\)