Surds and fractional indices · Revision guide

Learning objectives

For positive bases, a fractional index means root, then power.

a^{m/n}=\left(\sqrt[n]{a}\right)^m\quad(a>0;\ m\in\mathbb{Z},\ n\in\mathbb{Z}_{>0})

For a non-zero base and positive integer power, a negative index gives a reciprocal.

a^{-k}=\frac{1}{a^k}\quad(a\ne0,\ k\in\mathbb{Z}_{>0})

Take square factors outside the root. Collect only like surds.

\sqrt{ab}=\sqrt a\sqrt b\quad(a,b\ge0)

Rationalise a single surd denominator by multiplying both parts by that surd.

\frac{1}{\sqrt a}=\frac{\sqrt a}{a}\quad(a>0)

For a two-term surd denominator, multiply both parts by its non-zero conjugate.

Simplify the surd expression exactly.

\sqrt{50}+3\sqrt8

Extract the square factors.

\sqrt{50}=5\sqrt2,\qquad\sqrt8=2\sqrt2

5\sqrt2+3(2\sqrt2)=(5+6)\sqrt2

11\sqrt2

Evaluate the fractional power.

27^{2/3}=\left(\sqrt[3]{27}\right)^2=3^2

9

A root of a sum is not usually the sum of the roots. A negative index does not make the answer negative.