1.08 Evaluate expressions involving powers
We have now looked at exponents and how they can be written in exponential form or expanded form as a product of factors.
A base to the power of any other number, e.g. $3^4$34, can be read as "three to the power of four", and means that the base number is multiplied by itself the number of times shown in the power.
$3^4=3\times3\times3\times3$34=3×3×3×3
Now we would like to evaluate exponents by actually calculating the product of all of the factors.
The applet below allows us to select a base and an exponent, then see the evaluated product. Try varying the bases and exponents (by moving the sliders) to see how the numbers change.
- What is the largest number you can make?
- What is the smallest number you can make?
- Does the product change more if you slide just the base or just the exponent? Explain.
We may be asked to evaluate just a single power, for example $2^3=8$23=8, or we may be asked to evaluate an expression such as $2^3-3^2$23−32. When working with expression, we need to remember our order of operations.
Worked example
Question 1
Evaluate $9^2-2^3$92−23 by first rewriting the expression as a product of factors in expanded form.
Think: Exponents tell us how many times we are multiplying a number by itself. We have both exponents and subtraction, so we will need to remember our order of operations.
Do:
| $9^2-2^3$92−23 |
Original expression |
|
| $=$= | $9\times9-2\times2\times2$9×9−2×2×2 |
Rewriting in expanded form |
| $=$= | $81-8$81−8 |
Multiplying the factors |
| $=$= | $73$73 |
Subtracting |
Practice questions
QUESTION 2
Evaluate $2^6$26.
QUESTION 3
Evaluate $3^3\cdot3^2$33·32.
QUESTION 4
Evaluate $12^2-2^2$122−22 .