Rational Exponents and Real Roots

Algebra Foundation

Algebra 2

Explore advanced functions, complex numbers, polynomials, rational and radical expressions, exponentials, logarithms, conics, sequences, introductory trigonometry, probability, and statistics.

Learn with original explanations and worked examples. OpenStax Intermediate Algebra 2e was consulted for topic coverage.

Study every current unit through original concept lessons, formula derivations, experiments, quizzes, and practice sets.

Algebra 2 · Unit 5 · Topic 5.1

Rational Exponents and Real Roots

For a positive real base a, a^(m/n) = (ⁿ√a)ᵐ for integers m and positive integer n.

Definition and explanation

For a positive real base a, a^(m/n) = (ⁿ√a)ᵐ for integers m and positive integer n. Real even roots require nonnegative radicands; real odd roots can also have negative radicands. Negative exponents require nonzero bases.

Worked example

Evaluate 32^(2/5). Its fifth root is 2, so 32^(2/5) = 2² = 4. Also, the real cube root of −64 is −4 because (−4)³ = −64.

Common mistake

An even root of a negative real number is not real; do not apply positive-base exponent rules without checking the domain.

Key takeaway

For a positive real base a, a^(m/n) = (ⁿ√a)ᵐ for integers m and positive integer n.

Explain each step of the worked example, then check the units, signs, and conditions that apply.

Further reading

Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.