nur suci romadhon_12.84.202.034_5A2

8/17/2019 nur suci romadhon_12.84.202.034_5A2

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I n d i c e s o r P o w e r s

Fractional Powers

o far we have dealt with integer powers both positive and negative. What

would we do if we had a fraction for a power, like a1

2 . To see how to deal

with fractional powers consider the following:

Suppose we have two identical numbers multiplying together to give

another number, as in, for example

7×7=49

Then we know that 7 is a suare root of !". That is, if

72=49 then 7=√ 49

#ow suppose we found that

a p×a

p=a

That is, when we multiplied a p by itself we got the result a . This

means that a p must be a suare root of a .

$owever, look at this another way: noting that a=a1 , and also that, from

the first rule, a p

×a p

=a2 p

we see that if a p

×a p

=a then

a2 p=a1

from which

2 p=1

and so

p=1

2

This shows that a

1

2

must be the suare root ofa

. That is

S


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What do we mean by81

1

2 & 'or this we need to know what number

when multiplied by itself gives +(. The answer is ". So81

1

2=√ 81=9 .

Example

What about243

1

5 & What number when multiplied together five times

gives us *!& -f we are familiar with timestables we might spot that

243=3×81 , and also that 81=9×9 . So

24315=(3×9×9)

15=(3×3×3×3×3)

15

So multiplied by itself five times euals *!. $ence

243

1

5=3

#otice in doing this how important it is to be able to recognise what factors

numbers are made up of. 'or example, it is important to be able torecognise that:

16=24 , 16=42 , 81=92 , 81=34 , and so on.

/ou will find calculations much easier if you can recognise in numbers

their composition as powers of simple numbers such as *, , ! and 0.

1nce you have got these firmly fixed in your mind, this sort of calculation

becomes straightforward.

What happens if we take a3

4 &

We can write this as follows:

a

3

4=(a1

4 )3 using the second rule (am)n=amn

Example


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What do we mean by16

3

4 &

16

34=(16

14 )3

¿ (2 )3

¿8

2lternatively,

16

3

4=(24)3

4

¿ (2 )3

¿8

We can also think of this calculation performed in a slightly different way.

#ote that instead of writing (am)n=amn we could write (an)m=amn

because mn is the same as nm .

Example

What do we mean by 82

3 & 1ne way of calculating this is to write

8

2

3=(81

3 )2

¿ (2 )2

¿4

2lternatively,

8

2

3=(82)1

3

¿ (64)1

3

¿4

Additional Note

3oing this calculation the first way is usually easier as it reuires

recognising powers of smaller numbers.

'or example, it is straightforward to evaluate27

5

3 as


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27

5

3=(271

3 )5=35=243

because, at least with practice, you will know that the cube root of *7 is .

Whereas, evaluation in the following way

27

5

3=(275)1

3=143489071

3

would reuire knowledge of the cube root of 14348907 .

Writing these results down algebraically we have the following important

point:

Key Point

4oth results are exactly the same.

nur suci romadhon_12.84.202.034_5A2

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