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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

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).

  1. The denominator 3 means cube root.
  2. Cube root of 27 is 3.
  3. 3^2 = 9.

Answer: 9

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. Evaluate 81^(1/2).

Show solution
  1. Square root of 81 is 9.

Answer: 9

2. Evaluate 8^(2/3).

Show solution
  1. Cube root of 8 is 2, then square.

Answer: 4

3. Rewrite sqrt(x) as x to what exponent?

Show solution
  1. Square root means exponent 1/2.

Answer: 1/2

4. Evaluate 32^(1/5).

Show solution
  1. The fifth root of 32 is 2.

Answer: 2

5. Evaluate 64^(2/3).

Show solution
  1. Cube root of 64 is 4, then square.

Answer: 16

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