Rational Exponents
A free College Algebra lesson from the “Rational Exponents and Radicals” unit, with a worked example and practice problems including step-by-step solutions.
Rational exponents connect roots and powers. An exponent of 1/2 means square root, and an exponent of m/n means take the nth root and raise to the m power.
What you'll learn
- Rewrite radicals as exponents
- Evaluate rational exponents
- Use roots and powers together
Why it matters: Measurement scaling, growth formulas, and scientific models often use fractional powers because roots and powers happen together. The denominator names the root, while the numerator names the power.
Worked example
Problem. Evaluate 27^(2/3).
- The denominator 3 means cube root.
- Cube root of 27 is 3.
- 3^2 = 9.
Answer: 9
Practice problems
1. Evaluate 81^(1/2).
Show solution
- Warm-up: First identify exactly what the question is asking: Evaluate 81^(1/2).
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Square root of 81 is 9.
- Check the result by substituting or estimating: the response should match 9 and make sense in the original problem.
Answer: 9
2. Evaluate 8^(2/3).
Show solution
- Core Practice: First identify exactly what the question is asking: Evaluate 8^(2/3).
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Cube root of 8 is 2, then square.
- Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.
Answer: 4
3. Rewrite sqrt(x) as x to what exponent?
Show solution
- Challenge: First identify exactly what the question is asking: Rewrite sqrt(x) as x to what exponent?
- For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
- Square root means exponent 1/2.
- Check the result by substituting or estimating: the response should match 1/2 and make sense in the original problem.
Answer: 1/2
4. Evaluate 32^(1/5).
Show solution
- Roots: First identify exactly what the question is asking: Evaluate 32^(1/5).
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- The fifth root of 32 is 2.
- Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.
Answer: 2
5. Evaluate 64^(2/3).
Show solution
- Powers and Roots: First identify exactly what the question is asking: Evaluate 64^(2/3).
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Cube root of 64 is 4, then square.
- Check the result by substituting or estimating: the response should match 16 and make sense in the original problem.
Answer: 16
Practice this interactively with instant feedback and an AI tutor.
Practice Rational Exponents Take the free placement check