How to Solve Rational Exponents?

Rational exponents are exponents of numbers that are expressed as rational numbers, that is, in \(a^{\frac{p}{q}}\), \(a\) is the base, and \(\frac{p}{q}\) is the rational exponent where \(q ≠ 0\).

In rational exponents, the base must be a positive integer. Some examples of rational exponents are: \(2^{\frac{1}{3}}\), \(5^{\frac{5}{9}}\),\(10^{\frac{10}{3}}\).

Related Topics

• How to Solve Rational Exponents and Radicals

A step-by-step guide to rational exponents

Rational exponents are defined as exponents that can be expressed in the form of \(\frac{p}{q}\), where \(q≠0\).

The general symbol for rational exponents is \(x^{\frac{m}{n}}\), where \(x\) is the base (positive number) and \(\frac{m}{n}\) is a rational power. Rational exponents can also be written as \(x^{\frac{m}{n}}\) \(=\sqrt[n]{m}\)

• \(\color{blue}{x^{\frac{1}{n}}=\sqrt[n]{x}}\)
• \(\color{blue}{x^{\frac{m}{n}}=\:\left(\sqrt[n]{x}\right)^m\:or\:\sqrt[n]{\left(x^m\right)}}\)

Rational exponents formulas

Let’s review some of the formulas for rational exponents used to solve various algebraic problems. The formula for integer exponents is also true for rational exponents.

• \(\color{blue}{a^{\frac{m}{n}}\times a^{\frac{p}{q}}=a^{\left(\frac{m}{n}+\frac{p}{q}\right)}}\)
• \(\color{blue}{a^{\frac{m}{n}}\div a^{\frac{p}{q}}=a^{\left(\frac{m}{n}-\frac{p}{q}\right)}}\)
• \(\color{blue}{a^{\frac{m}{n}}\times b^{\frac{m}{n}}=\left(ab\right)^{\frac{m}{n}}}\)
• \(\color{blue}{a^{\frac{m}{n}}\div b^{\frac{m}{n}}=\left(a\div b\right)^{\frac{m}{n}}}\)
• \(\color{blue}{a^{-\frac{m}{n}}=\left(\frac{1}{a}\right)^{\frac{m}{n}}}\)
• \(\color{blue}{a^{\frac{0}{n}}=a^0=1}\)
• \(\color{blue}{\left(a^{\frac{m}{n}}\right)^{\frac{p}{q}}=a^{\frac{m}{n}\times \frac{p}{q}}}\)
• \(\color{blue}{x^{\frac{m}{n}}=y⇔x=y^{\frac{n}{m}}}\)

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Rational Exponents – Example 1:

solve. \(8^{\frac{1}{2}}\times 8^{\frac{1}{2}}\)

Use this formula to solve rational exponents: \(\color{blue}{a^{\frac{m}{n}}\times a^{\frac{p}{q}}=a^{\left(\frac{m}{n}+\frac{p}{q}\right)}}\)

\(8^{\frac{1}{2}}\times 8^{\frac{1}{2}}\) \(=8^{\left(\frac{1}{2}+\frac{1}{2}\right)}\)

\(=8^{1}=8\)

Rational Exponents – Example 2:

Solve. \(2^{\frac{1}{4}}\times 7^{\frac{1}{4}}\)

Use this formula to solve rational exponents: \(\color{blue}{a^{\frac{m}{n}}\times b^{\frac{m}{n}}=\left(ab\right)^{\frac{m}{n}}}\)

\(2^{\frac{1}{4}}\times 7^{\frac{1}{4}}\) \(=(2\times7)^{\frac{1}{4}}\)

\(=14^{\frac{1}{4}}\)

Exercise for Rational Exponents

Evaluate the following rational exponents.

1. \(\color{blue}{25^{\frac{1}{2}}}\)
2. \(\color{blue}{81^{\frac{5}{4}}}\)
3. \(\color{blue}{(2x^{\frac{2}{3}})(7x^{\frac{5}{4}})}\)
4. \(\color{blue}{8^{\frac{1}{2}}\div 8^{\frac{1}{6}}}\)
5. \(\color{blue}{(\frac{16}{9})^{-\frac{1}{2}}}\)
6. \(\color{blue}{\left(8x\right)^{\frac{1}{3}}\left(14x^{\frac{6}{5}}\right)}\)

1. \(\color{blue}{5}\)
2. \(\color{blue}{243}\)
3. \(\color{blue}{14x^{\frac{23}{12}}}\)
4. \(\color{blue}{2}\)
5. \(\color{blue}{\frac{3}{4}}\)
6. \(\color{blue}{28x^{\frac{23}{15}}}\)