# 2.1 Multiplying powers with the same base number

At the beginning of the week the first example used was the width of the observable universe, which is about 92 billion light years, where each light year is about 9.5 trillion kilometres.

So, to work this out in kilometres, multiply 92 billion by 9.5 trillion. Both these numbers can be shown in scientific notation as follows:

So the calculation becomes:

This still leaves what looks like a complicated calculation. However, there are some handy short cuts that can be used when dealing with powers with the same base number (in this case 10 is the base number). To explore this, here is a simpler example: .

Writing these numbers out in full, the calculation is:

1 000 000 written in powers of 10 is 10^{6}.

So, .

Can you spot anything that relates the powers of ten in the answer to those in the original numbers?

The powers of 10 in the original numbers were 2 and 4, and in the answer 6. It would seem that if you add the powers of 10 in the original numbers, you arrive at the power of 10 in the answer. So:

This rule can be used whenever you are dealing with multiplication of numbers with the same base number and gives a quick way to calculate the width of the observable universe. Note that the base number remains the same! You can do this in the next activity after you’ve had a go at some other examples.

## Activity 3 Multiplying powers with the same base number

Work out the following, giving your answer first in power form before calculating the answer to the sum. You can use a calculator to work these out for parts b) to d).

- a.

### Comment

These numbers both have the same base number (10), so add the powers for the final answer.

### Answer

a.

- b.

### Comment

The base number in this case is 2.

### Answer

- b.

- c.

### Answer

- c.

- d.Now work out the width of the universe. Remember that the universe is 92 billion light years across and a light year is about 9.5 trillion kilometres.

### Answer

- d.
First express both numbers in scientific notation:

Our first number is not between 1 and 10, so this is not yet shown in correct scientific notation.

That means the width of the observable universe is about, 8.74 x 10^{23} km and you’ve just worked out the answer to the original problem – well done!

You might reasonably be asking yourself if there is a similar rule for dividing numbers with the same base as there is for multiplying them. Continue to the next section to find out now.