We have looked at expressions involving negative exponents and how we can express them as a fraction with a positive exponent. Recall, that the negative exponent law states:
$a^{-n}=\frac{1}{a^n}$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 exponential 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 exponent. How do we apply the negative exponent rule to this expression? We know that we want to take the reciprocal of the base, and rewrite it with a positive exponent. 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 exponent laws, we were able to take an expression involving a negative exponent, and re-write it the much more familiar form of a fraction with a positive numerator and denominator. We could then evaluate the positive exponent term
Examples
Question 1
Evaluate $2^{-6}$2−6 by first expressing it with a positive exponent. Give your answer as a fraction.
Question 2
Evaluate $3^{-4}+3^{-1}$3−4+3−1 by using exponent 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.