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Rational Exponents

A free Algebra II lesson from the “Rational Exponents and Radicals” unit, with a worked example and practice problems including step-by-step solutions.

Rational exponents connect powers and roots. An exponent of 1/2 means square root, 1/3 means cube root, and m/n means take the nth root and raise to the m power.

What you'll learn

Why it matters: Power-spectrum frequency math, growth-rate models, and engineering yield calculations all use fractional exponents. The form x^(m/n) is the same as the n-th root of x^m, which is the bridge between exponent notation and radical notation.

Worked example

Problem. Evaluate 16^(1/2).

  1. An exponent of 1/2 means square root.
  2. sqrt(16) = 4.
  3. So 16^(1/2) = 4.

Answer: 4

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 25^(1/2).

Show solution
  1. Square root of 25 is 5.

Answer: 5

2. Evaluate 27^(1/3).

Show solution
  1. Cube root of 27 is 3.

Answer: 3

3. Evaluate 8^(2/3).

Show solution
  1. Cube root of 8 is 2.
  2. 2^2 = 4.

Answer: 4

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

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

Answer: 1/2

5. 64^(1/3) equals...

Choices: 4 · 8 · 16 · 32

Show solution
  1. 4^3 = 64.

Answer: 4

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