AHL 1.10 (HL)—Rational exponents

A rational exponent is defined so that a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ when the expression is real. The denominator n determines the root; the numerator m determines the power.

8^(2/3) = (∛8)² = 2² = 4. Rewriting the expression as a root first makes the order of operations visible and avoids treating m/n as an ordinary multiplier.

For even n, a negative base may not produce a real value. Keep the domain in mind, and remember that a^(−m/n) is the reciprocal of a^(m/n), not a negative result.