UK Secondary (7-11)
Powers
Lesson

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.

Did you know?

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.

1. First let's convert $4^3$43 to expanded form.

2. 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