We have looked at expressions involving negative indices and how we can express them as a fraction with a positive index. Recall, that the negative index law states:

The negative index law

$a^{-n}=\frac{1}{a^n}$a−n=1an, where $a\ne0$a≠0.

Following this law, we saw how an expression such as $2^{-3}$2−3 can be rewriten in positive index form by taking the reciprocal: $2^{-3}=\frac{1}{2^3}$2−3=123.

We know $\frac{1}{2^3}=\frac{1}{8}$123=18. So we have now have three was to represent the fraction $\frac{1}{8}$18.

In this chapter, we will extend this concept and evaluate expressions involving negative powers.

Let's look at an example to see this in action.

Worked Example

Express $5\times2^{-6}$5×2−6 as a fraction in simplest form.

Think: To write it as a fraction, we need to first remove the negative index. How do we apply the negative index rule to this expression? We know that we want to take the reciprocal of the base, and rewrite it with a positive index. We can then multiply this by $5$5. Finally, we can evaluate the $2^6$26 in the denominator.

Do:

$5\times2^{-6}$5×2−6

$=$=

$5\times\frac{1}{2^6}$5×126

$=$=

$\frac{5}{2^6}$526

$=$=

$\frac{5}{64}$564

Reflect:By using the index laws, we were able to take an expression involving a negative index, and re-write it the much more familiar form of a fraction with a positive numerator and denominator. We could then evaluate the positive index term

Examples

Question 1

Evaluate $2^{-6}$2−6 by first expressing it with a positive index. Give your answer as a fraction.

Question 2

Evaluate $3^{-4}+3^{-1}$3−4+3−1 by using index laws, giving your answer as a simplified fraction.

Question 3

Evaluate $2^{-3}\times4^{-2}$2−3×4−2 without the use of a calculator.