Negative and fractional powers
Introduction
Sometimes it is useful to use negative and fractional powers.
These are explained on this leaflet.
1. Negative powers
Sometimes you will meet a number raised to a negative power.
This is interpreted as follows:
![](./articles_imgs/862/pic1.GIF)
This can be rearranged into the alternative form:
![](./articles_imgs/862/pic2.GIF)
Example
![](./articles_imgs/862/pic3.GIF)
Exercises
1. Write the following using only positive powers:
![](./articles_imgs/862/pic4.GIF)
2. Without using a calculator evaluate ![](./articles_imgs/862/pic5.GIF)
Answers
![](./articles_imgs/862/pic6.GIF)
2. Fractional powers
To understand fractional powers you first need to have an
understanding of roots, and in particular square roots and cube
roots.
When a number is raised to a fractional power this is
interpreted as follows:
![](./articles_imgs/862/pic7.GIF)
So,
is a square root of a
is the cube root of a
is a fourth root of a
Example
![](./articles_imgs/862/pic8.GIF)
Fractional powers are useful when we need to calculate roots
using a scientific calculator. For example to find we
rewrite this as which can be evaluated using a scientific
calculator. You may need to check your calculator manual to find
the precise way of doing this, probably with the buttons or ![](./articles_imgs/862/pic12.GIF)
Check that you are using your calculator correctly by
confirming that = 1.6814 (4 dp)
More generally means or
equivalently ![](./articles_imgs/862/pic15.GIF)
![](./articles_imgs/862/pic16.GIF)
Example
![](./articles_imgs/862/pic17.GIF)
Exercises
1. Use a calculator to find ![](./articles_imgs/862/pic18.GIF)
2. Without using a calculator, evaluate ![](./articles_imgs/862/pic19.GIF)
3. Use the third law of indices to show that
![](./articles_imgs/862/pic20.GIF)
and equivalently
![](./articles_imgs/862/pic21.GIF)
Answers
![](./articles_imgs/862/pic22.GIF)
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