Succeed with maths – Part 2
Succeed with maths – Part 2

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Succeed with maths – Part 2

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:

92 billion equation left hand side equals right hand side 9.2 multiplication 10 super 10

9.5 trillion equation left hand side equals right hand side 9.5 multiplication 10 super 12

So the calculation becomes:

9.2 multiplication 10 super 10 multiplication 9.5 multiplication 10 super 12

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: 10 squared multiplication 10 super four.

Writing these numbers out in full, the calculation is:

multiline equation row 1 10 squared multiplication 10 super four equals open 10 multiplication 10 close multiplication open 10 multiplication 10 multiplication 10 multiplication 10 close row 2 equals equation left hand side 100 multiplication 10 times 000 equals right hand side one times 000 times 000

1 000 000 written in powers of 10 is 106.

So, equation left hand side 10 squared multiplication 10 super four equals right hand side 10 super six.

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: equation sequence 10 squared multiplication 10 super four equals 10 super open two plus four close equals 10 super six

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 _unit6.2.2 Activity 3 Multiplying powers with the same base number

Timing: Allow approximately 10 minutes

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.10 super four multiplication 10 cubed

Discussion

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

Answer

  • a.equation sequence 10 super four multiplication 10 cubed equals 10 super open four plus three close equals 10 super seven equals 10 000 000

    10 super seven equals 10 000 000

  • b.two squared multiplication two super four

Discussion

The base number in this case is 2.

Answer

  • b.multiline equation line 1 equation sequence two squared multiplication two super four equals two super open two plus four close equals two super six line 2 two super six equals 64
  • c.open negative three close squared multiplication open negative three close cubed

Answer

  • c.multiline equation line 1 equation sequence open negative three close squared multiplication open negative three close cubed equals open negative three close super open two plus three close equals open negative three close super five line 2 equation left hand side open negative three close super five equals right hand side negative 243
  • 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.Width of the universe equals 9.2 billion multiplication 9.5 trillion km

    First express both numbers in scientific notation:

    multiline equation line 1 92 billion equation left hand side equals right hand side 9.2 multiplication 10 super 10 line 2 9.5 trillion equation left hand side equals right hand side 9.5 multiplication 10 super 12

    multiline equation row 1 So the width equals 9.2 multiplication 10 super 10 multiplication 9.5 multiplication 10 super 12 km equation left hand side equals right hand side 9.2 multiplication 9.5 multiplication 10 super 10 multiplication 10 super 12 km row 2 equals 87.4 multiplication 10 super 22 km

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

    87.4 equals 8.74 multiplication 10 super one

    multiline equation row 1 So comma 87.4 multiplication 10 super 22 equals 8.74 multiplication 10 super one multiplication 10 super 22 row 2 equals 8.74 multiplication 10 super 23

That means the width of the observable universe is about, 8.74 x 1023 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.

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