For a positive base a and positive integers m,n, a 1n= [n]a and a mn=([n]a)m. The denominator tells you the root; the numerator tells you the power. Root first is often easier. Example 81 34=([4]81)3=33=27. 32- 25= 1([5]32)2= 14. [4]81=3 is the positive fourth root. Likewise 16 12=4, not +-4. The equation x2=16 has two solutions, which is a different request. The positive-base condition keeps these general fractional-index laws unambiguous. Use the ordinary cube-root meaning for [3]-8=-2; do not apply every fractional-power manipulation indiscriminately to negative bases.

For a positive base \(a\) and positive integers \(m,n\), \(a^{\frac1n}=\sqrt[n]{a}\) and \(a^{\frac mn}=(\sqrt[n]{a})^m\). The denominator tells you the root; the numerator tells you the power. Root first is often easier.

\(81^{\frac34}=(\sqrt[4]{81})^3=3^3=27\).

\(32^{-\frac25}=\dfrac1{(\sqrt[5]{32})^2}=\dfrac14\).

\(\sqrt[4]{81}=3\) is the positive fourth root. Likewise \(16^{\frac12}=4\), not \(\pm4\). The equation \(x^2=16\) has two solutions, which is a different request.

The positive-base condition keeps these general fractional-index laws unambiguous. Use the ordinary cube-root meaning for \(\sqrt[3]{-8}=-2\); do not apply every fractional-power manipulation indiscriminately to negative bases.

Fractional indices and roots

Fractional indices and roots