How to Solve Rational Exponents?

An exponential expression (also called fractional exponents) of the form \(a^m\) has a rational exponent if \(m\) is a rational number (as opposed to integers). Here, you learn more about solving rational exponents problems.

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: \(2^{\frac{1}{3}}\), \(5^{\frac{5}{9}}\),\(10^{\frac{10}{3}}\).

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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.

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

Rational Exponents – Example 1:

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

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

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


Rational Exponents – Example 2:

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

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

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


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. \(\color{blue}{\left(8x\right)^{\frac{1}{3}}\left(14x^{\frac{6}{5}}\right)}\)
This image has an empty alt attribute; its file name is answers.png
  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. \(\color{blue}{28x^{\frac{23}{15}}}\)

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