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Powers and roots

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Multiplication

How can we work out 23 × 25?

23 = 2 × 2 × 2

25 = 2 × 2 × 2 × 2 × 2

so 23 × 25 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 28

There are 3 twos from 23 and 5 twos from 25, so altogether there are 8 twos.

In general, 2m × 2n =2(m + n)

Examples

25 × 24 = 2(5 + 4) = 29

27 × 23 = 2(7 + 3) = 210

The rule also works for other numbers, so

34 × 32 = 3(4 + 2) = 36

256 × 254 = 25(6 + 4) = 2510

Division

If you divide 25 by 23 you see that some of the 2's cancel:

Dividing numbers with roots or powers

shows five 2s divided by three 2s.

Five 2s are divided by three 2s

So 25 ÷ 23 = 22

In general, 2m ÷ 2n = 2(m - n)

Example

25 ÷ 22 = 2(5 - 2) = 23

27 ÷ 23 = 2(7 - 3) = 24

The rule also works for other numbers, so

510 ÷ 53 =5(10 - 3) = 57

459 ÷ 454 = 45(9 - 4) = 455

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