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.

Task!

Write each of the following using a positive index. Use Key Point 9.

$\frac{1}{t^{- 4}}$ ,
$1 7^{- 3}$ ,
$y^{- 1}$ ,
$1 0^{- 2}$

$t^{4}$

$\frac{1}{1 7^{3}}$

$\frac{1}{y}$

$\frac{1}{1 0^{2}} which equals \frac{1}{100} or 0.01$

Task!

Simplify $\frac{a^{8} \times a^{7}}{a^{4}}$

Use the first law of indices to simplify the numerator:

$\frac{a^{15}}{a^{4}}$ Now use the second law to simplify the result:

$a^{11}$

Task!

Simplify $\frac{m^{9} \times m^{- 2}}{m^{- 3}}$

First simplify the numerator using the first law of indices:

$\frac{m^{7}}{m^{- 3}}$ Then use the second law to simplify the result:

$m^{7 - (- 3)} = m^{10}$

Exercises

Write the following numbers using a positive index and also express your answers as decimal fractions:
1. $1 0^{- 1}$ ,
2. $1 0^{- 3}$ ,
3. $1 0^{- 4}$
Simplify as much as possible:
1. $x^{3} x^{- 2}$ ,
2. $\frac{t^{4}}{t^{- 3}}$ ,
3. $\frac{y^{- 2}}{y^{- 6}}$ .

1. $\frac{1}{10} = 0.1$ ,
2. $\frac{1}{1 0^{3}} = 0.001$ ,
3. $\frac{1}{1 0^{4}} = 0.0001$ .
1. $x^{1} = x$ ,
2. $t^{4 + 3} = t^{7}$ ,
3. $y^{- 2 + 6} = y^{4}$ .