Mathematics for science and technology
Mathematics for science and technology

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Mathematics for science and technology

1.5 The power of zero

At first sight a power of zero wouldn’t seem to mean anything. However, by using Rule 2 (dividing with powers) you can show that anything to the power zero is 1.

Clearly 2 ÷ 2 = 1.

However, using Rule 3, and the fact that 2 = 21, you can re-write this as:

21 ÷ 21 = 2(1–1) = 20

Showing that 20 = 1

This can be generalised to give: a0 = 1.

Note that you can now explain why equation left hand side a super negative n equals right hand side one divided by a super n.

By writing an as a(0 − n)

and, using Rule 2, this is the same as a0 ÷ an.

Since a0 = 1 this is then the same as:

equation left hand side one division a super n equals right hand side one divided by a super n

In the same way, you can show that

equation left hand side a super n equals right hand side one divided by a super negative n

one divided by a super negative ncan be written as 1 ÷ an

So, using Rule 2 and a0 = 1 gives:

multiline equation row 1 one divided by a super negative n equals multiline equation line 1 a super zero division a super negative n row 2 equals a super open zero minus open negative n close close row 3 equals a super n

Now try some examples.

Activity 1 Working with power rules

Timing: Allow about 15 minutes

Simplify each of the following examples. For a) and b) do this without a calculator initially.

  • a.four super three divided by two
  • b.125 super negative two divided by three
  • c.48 times a super eight division six times a super four
  • d.three times a cubed division 12 times a super nine
  • e.a super nine multiplication a cubed divided by a super six
  • f.open 16 times a super six divided by nine times a super four close super one divided by two

Discussion

  • a.equation left hand side four super three divided by two equals right hand side open four super one divided by two close cubed which can be written as multiline equation row 1 open Square root of four postfix times close super three equals two cubed row 2 equals eight
  • b.
  • c.multiline equation row 1 48 times a super eight division six times a super four equals eight times a super open eight minus four close row 2 equals multiline equation line 1 eight times a super four
  • d.multiline equation row 1 three times a cubed division 12 times a super nine equals multiline equation line 1 a super open three minus nine close divided by four row 2 equals multiline equation line 1 multiline equation line 1 a super negative six divided by four row 3 equals one divided by multiline equation line 1 multiline equation line 1 four times a super six
  • e.multiline equation row 1 a super nine multiplication a cubed divided by a super six equals a super open nine plus three close division a super six row 2 equals a super open nine plus three minus six close row 3 equals a super six
  • f.multiline equation row 1 multiline equation line 1 open 16 times a super open six minus four close divided by nine close super one divided by two equals multiline equation line 1 open multiline equation line 1 16 times a squared divided by nine close super one divided by two row 2 equals four times a divided by three

Although you may arrive at the answer to an activity in a different way from that given here (because the rules of indices can often be applied in a different order), you should always get the same final answer.

All the examples you have looked at in this section have a simple answer but this will not always be the case, particularly where the calculation relates to a situation modelled on the real world.

In the next section you will be able to put these ideas in to practice again, as you learn about how to use powers to represent very small and very large numbers.

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