Powers
Yes, exponents are the little numbers written up and next to other numbers, the $3$3 in the picture above.
Exponents are used simply to indicate that a number is multiplied by itself many times.
For example, $2^3$23 is just another way of writing $2\times2\times2$2×2×2.
You may be thinking to yourself how unnecessary exponents are; after all, it's not hard to write $2\times2\times2$2×2×2. But suppose you want to write ten $2$2's multiplied together. You could either write $2\times2\times2\times2\times2\times2\times2\times2\times2\times2$2×2×2×2×2×2×2×2×2×2 or you could simply write $2^{10}$210. Now suppose you want to write a million $2$2's multiplied together... I'm sure you get the picture.
Exponential notation (the notation that involves exponents) is actually very straightforward. Consider $2^3$23. Here the exponent, $3$3, tells us how many times the base, $2$2, should be used in the multiplication. The following demonstration illustrates more of this notation. Try varying the bases and exponents (by moving the sliders) to see how the numbers change.
A common way to describe exponents is like this:
- $a^m$am means $a$a multiplied $m$m times
- for example, $3^4=3$34=3 multiplied $4$4 times = $3\times3\times3\times3$3×3×3×3
If you find that confusing, you can also think of it this way
- use $a$a, $m$m times in the multiplication
-
for example, $3^4$34 = use the number $3$3, four times in the multiplication = $3\times3\times3\times3$3×3×3×3
Examples
Now working out the value of expressions that have exponents is done by remembering what the notation actually means.
Question 1
We want to evaluate $4^3$43 using repeated multiplication. First let's convert $4^3$43 to expanded form. Using your answer to part (a), state the value of $4^3$43.
Here's another one
Question 2
Evaluate $2^3\times2^4$23×24
Question 3
Write this expression using index notation, in simplest form.
$5\times5$5×5