Rational Exponents
In short: Rational exponents are fractional powers such as 27^(2/3), where the denominator names a root and the numerator names a power, so \(a^{m/n} = (\sqrt[n]{a})^m\).
Algebra 1Numbers and OperationsN-RN.A.1
What is Rational Exponents?
Powers need not be whole numbers, and rational exponents put a fraction in the power spot so that roots and powers share one notation: \(a^{1/n}\) means the \(n\)th root of \(a\).
With a numerator other than 1, \(a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}\). The bottom number picks the root, the top number picks the power. Doing the root first keeps the arithmetic small, and every ordinary exponent law still holds.
How to spot it
- Rational exponents appear as a fraction up in the power spot, like \(16^{3/4}\) or \(x^{1/2}\).
- A problem asks you to rewrite a radical expression with an exponent, or the reverse.
- A formula involves a cube root or square root of a volume or area, such as a side length from a volume.
- A negative fraction in the exponent suggests a reciprocal as well as a root.
The method
- Read the fraction \(\frac{m}{n}\) in the exponent: n gives the root index and m gives the power.
- Take the n-th root of the base first when it is a perfect power.
- Raise that result to the power m.
- If the exponent is negative, take the reciprocal of the base first, then apply the steps.
Rational Exponents: solved example
Problem
A cube-shaped storage box has a volume of 1,728 cubic centimeters. Its edge is \(V^{1/3}\) long and each square side has area \(V^{2/3}\). Compute both.
Solution
Edge: \(1{,}728^{1/3} = \sqrt[3]{1{,}728} = 12\), since \(12^3 = 1{,}728\).
Face area: \(1{,}728^{2/3} = (12)^2 = 144\). The edge is 12 cm and one face has area 144 cm².
Try it
Question 1
Evaluate \(64^{5/6}\).
Show the answer
The 6th root of 64 is 2, and \(2^5 = 32\).
Question 2
Evaluate \(25^{-1/2}\).
Show the answer
\(25^{-1/2} = \frac{1}{25^{1/2}} = \frac{1}{5}\).
Rational Exponents vs. Exponent Rules
The general power rules explain how exponents combine, such as adding when multiplying like bases. Fractional powers are a special case of those rules: they say that an exponent of one over n is a root. Learn the rule set first, then see this notation as a compact way to write radicals.
Keep going
Lessons
Worksheets and quizzes
- Exponent Rules and Radicals – Worksheet
- Exponent Rules and Radicals – Quiz
- Radicals and Rational Exponents – Worksheet
- Radicals and Rational Exponents – Quiz
Related words
Whole dictionary: A–Z · Algebra 1 · Numbers and Operations