# 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.

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

## 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 $$x^{\frac{m}{n}}$$ $$=\sqrt[n]{m}$$

### 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: $$\color{blue}{a^{\frac{m}{n}}\times a^{\frac{p}{q}}=a^{\left(\frac{m}{n}+\frac{p}{q}\right)}}$$

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

$$=8^{1}=8$$

### 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. $$\color{blue}{28x^{\frac{23}{15}}}$$

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