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. \(\color{blue}{28x^{\frac{23}{15}}}\)

Recommended EffortlessMath Books

For full coverage of exponents, radicals, and rational exponent rules, the Algebra II for Beginners walks through each topic with worked examples and practice sets. For students preparing for college math, the Pre-Calculus for Beginners reviews rational exponents alongside logarithms and exponential functions.

Frequently Asked Questions

How do you solve rational exponents step by step?

Read the fraction in the exponent: the denominator is the root and the numerator is the power. Take the root first, then raise to the power, which keeps the numbers small. For 27^(2/3), the cube root of 27 is 3, and 3² = 9. If the exponent is negative, finish by taking the reciprocal.

How do you simplify rational exponents?

Use the usual exponent rules with fractions. When multiplying the same base, add the exponents: x^(1/2) · x^(1/4) = x^(3/4). When dividing, subtract them, and for a power of a power, multiply them: (x^(2/3))^6 = x⁴. Write the final answer with a positive exponent or as a radical, depending on what your teacher asks for.

How do you solve negative rational exponents?

A negative exponent means take the reciprocal, so x^(−m/n) = 1 ÷ x^(m/n). For 16^(−3/4), first find 16^(3/4): the fourth root of 16 is 2, and 2³ = 8. Then flip it to get 1/8. The negative sign does not make the answer negative; it only moves the base to the denominator.

How do you solve fractional exponents without a calculator?

Look for a base that is a perfect square, cube or other power that matches the denominator. For 32^(3/5), notice that 32 = 2⁵, so the fifth root of 32 is 2, and 2³ = 8. Rewriting the base as a power, then multiplying the exponents, turns most textbook problems into quick mental math.

How do you solve rational exponents with variables?

To solve an equation like x^(2/3) = 9, raise both sides to the reciprocal power, 3/2. That gives x = 9^(3/2) = (√9)³ = 27. Because the exponent has an even numerator, −27 also works, since (−27)^(2/3) = 9, so always check your solutions in the original equation.

How do you simplify rational exponents and radicals?

Switch between the two forms using x^(m/n) = ⁿ√(xᵐ). Radicals are easier to read, while exponents are easier to combine. For example, √x · ∛x becomes x^(1/2) · x^(1/3) = x^(5/6), which you can write back as the sixth root of x⁵.

Related EffortlessMath Lessons

If a topic on this page feels rusty, these short lessons go deeper:

• Rules of exponents
• How to simplify radicals
• How to solve radical equations
• How to multiply radicals
• How to convert between radical and rational exponent form