# 10.1.3 Multiplying and Dividing Numbers with the Same Base Number

How does this notation help in working out the width of the universe?

Consider what happens when you multiply two different powers of 10 together, say .

There are six 10s multiplied together, so we can write this as . So,

Have you noticed anything about the powers of ten in the answer and the original numbers?

You can get the same result by adding the powers together:

Try working out the “long way” and then by adding the powers together. Do your answers agree? The following rule applies in general: To multiply two numbers with the same base number, add the powers.

## Activity: Multiplying Numbers with the Same Base Number

Work out the following, giving your answer first in power form and then in decimal form.

(a)

(a)

(b)

(b)

(c)

(c)

## Activity: The Width of the Universe

You can 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 kilometers.

Width of the universe = km.

. First express both numbers in scientific notation: .

So the width = = . Multiplying 9.2 by 9.5 gives 87.4. Then adding the powers gives .

So the calculation is:

That means the width of the universe is .

We have a rule for multiplying two numbers with the same base number, but what about division?

Consider . This can be written as a fraction, and then canceled, as follows:

Therefore, .

This time you can get the same result by subtracting the powers:

Try working out the “long way,” and then by subtracting the powers. Are you convinced that the following rule works?

To divide two numbers with the same base number, subtract the powers.

Make a note of these rules about powers with the same base in your math notebook - remembering to highlight that they apply only with the same base number.

## Activity: Dividing Numbers with the Same Base Number

Without using your calculator, work out each of the following.

(a)

(a)

(b)

(b)

(c)