Negative indices

4 Negative indices

Sometimes a number is raised to a negative power. This is interpreted as follows:

Key Point 9

Negative Powers

a^{- m} = \frac{1}{a^{m}}, a^{m} = \frac{1}{a^{- m}}

Thus a negative index can be used to indicate a reciprocal.

Example 21

Write each of the following expressions using a positive index and simplify if possible.

$2^{- 3}$ ,
$\frac{1}{4^{- 3}}$ ,
$x^{- 1}$ ,
$x^{- 2}$ ,
$1 0^{- 1}$

Solution

$2^{- 3} = \frac{1}{2^{3}} = \frac{1}{8}$ ,
$\frac{1}{4^{- 3}} = 4^{3} = 64$ ,
$x^{- 1} = \frac{1}{x^{1}} = \frac{1}{x}$ ,
$x^{- 2} = \frac{1}{x^{2}}$ ,
$1 0^{- 1} = \frac{1}{1 0^{1}} = \frac{1}{10}$ or 0.1.

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