## 3. Multiplying and dividing using powers

You can perform fairly daunting calculations by using a simple property of powers. First, look at a very simple sum like:

100 × 1 000 000

It's obvious that the answer is 100 000 000 (one hundred million), but now look at this when we write the numbers as powers:

10^{2} × 10^{6} = 10^{8} = 10^{(6+2)}

You will have spotted that the superscript for the answer (8) is the sum of the two superscripts of the multipliers (2 and 6). Thus we can multiply quantities expressed as exact powers of the same base by adding their exponents. Similarly, if we want to divide, we subtract the powers:

1 000 000 / 100 = 10^{6} / 10^{2} = 10^{(6-2)} = 10^{4} = 10 000

Easy with exact multiples of ten, but how is this used with 'real' numbers? You have already seen that you can express a number as a multiplier and a power of ten (commonly termed 'scientific notation'). So that:

3756 = 3.756 × 10^{3} and 696 = 6.96 × 10^{2}

Using the same rule as we applied to exact powers of ten, we can multiply these two numbers:

3756 × 696 = 3.756 × 10^{3} × 6.96 × 10^{2}

Again, we can add the powers of ten to make one multiplier:

3.756 × 10^{3} × 6.96 × 10^{2} = 3.756 × 6.96 × 10^{5}

Then we multiply the other two terms:

3.756 × 6.96 × 10^{5} = 26.204 × 10^{5} (= 2.6204 × 10^{6})

Perhaps not all that exciting in itself, but it does help with doing mental arithmetic approximations. Taking the same sum and rounding up the multipliers:

3.75 × 7 = 21 + 5 = 26
and 10^{3} × 10^{2} = 10^{5},

then: 3756 × 696 ~ 26 × 10^{5}, which is pretty close to the real answer!