Fractional and negative indices

A fractional index means a root: a^(1/n) is the nth root of a. a^(m/n) means take the nth root, then raise to the power m. So 8^(2/3) = (∛8)² = 2² = 4.

A negative index means one over: a⁻ⁿ = 1/aⁿ. Combining both, 27^(−2/3) = 1/27^(2/3) = 1/(∛27)² = 1/9. With letters, (16a⁴)^(−1/2) = 1/(4a²).

These rules let you write roots as powers, which is the first step in calculus. For example, √x = x^(1/2) and 1/x² = x⁻², so 3/√x can be written 3x^(−1/2). Writing every term as a power of x makes expressions easy to simplify, compare and later differentiate.