Powers and roots
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GCSE Maths

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Rational and Irrational Numbers

There are two different kinds of numbers, rational and irrational. The difference is that rational can be turned into fractions while irrational can't. Simple as that. There are a few traps of course...

Examples of rationals:

1 -88 2

Examples of irrationals:

  • Any form of pi is always irrational: , ,

(except of course which natually equals 1.)

  • Most square roots are irrational. For example:

However, look at ... As you'll know from your vast knowlege of Maths, the square root of 4 is 2. The rule is, all square roots, except for squared numbers (pretty obvious really).

The name for a number with a square root in it, e.g. 2 is a surd. We'll be coming to those in a little while...

Recurring decimals

So how exactly does something like become a fraction, you might wonder. There's actually a really neat method that you'll either understand, or just do.

1. Let's call f.

2. So 10f would equal (x 10 =)

3. If we take f from 10f we remove the fractionand we're left with 9f. (-= 8 )

4. Now just divide 9f by 9 to get f as a fraction. (8 9 =)



The "just learn it" method is like this:

= , = , = , etc...

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Surds is just a posh - and slighly obscure - word for numbers that have square roots in them, such as the example above: 2. However, it's vital you know how to manipulate them, because the exam board love throwing in questions on the non-calculator paper where you leave your answer in surd form.

Say you work something out and get the answer:

You'll lose a mark if you leave it at that though. So what can you do with it? Obviously you need to remove the divide 2. First however, you need to sort out . The best way to do this is run through all the square numbers and find which are factors of 44. In this case, the only one is: 4. We can then rewrite as:

Now, change that to 2, so you'll have: 2 x 2 x on the top, or simply:

The square root isn't effected by the divide now, so it's just a case of dividing 4 by 2 to get two. And we're left with:


More cool surd rules: (eek, did I just say "cool"??)

  1. =
  2. + Cannot be done!
  3. (a+)² = (a+)(a+) = a² + 2a+ b
  4. Express in the form where a and b are whole numbers. To do this, first, we need to remove the . Because 1 is just we can times the number by and get:. a = 5 and b = 2.

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Powers and roots

Of course you know what powers and roots are without explanation (squared and square roots for example). To use them, though, there are a few rules you need to straighten out:

  1. Mutiplying means adding the powers. E.g. 3³ x 3³ = 3 (notice the 3 remains the same.)
  2. And so dividing means taking away the powers. E.g. 3³ 3² = 3¹.
  3. Raising by a power means multiplying. E.g. (3³)³ = 3.
  4. Anything to the power of 1 is just itself.
  5. Anything to the power of 0 is 1.


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Rounded Off Values

When we are given a measurement, just how accurate is it, really? If we have 1.3 that number could have been rounded up, or down. This means it could have been between 1.25 and 1.34. (It can't be 1.35 or that would be rounded up to 1.4).

However, the exam board aren't so sophisticated, so the general (untruthful) rule is:

For 1.3 the real value could be between 1.25 and 1.35.

The same applies for any number. 6.653 would have a range of between 6.6525 and 6.6535.

Once you've found your upper and lower boundaries, you can find out the maximum and minimum values of a calculation.

Say you're given a rectangle:

To work out the minimum area, we need to times together both lower boundaries. That would be:

6.65 m x 13.35 m

which equals 88.7775 m² (note you DON'T round that figure).

Maximum would be: 6.75 m x 13.45 m = 90.7875 m²

However, it's not always that simple. Take the example:

To get the maximum here, you need the lower boundary of o and the upper boundary of n.

It's best, if you're unsure, just to try out various combinations, and see which gets you the result you need.

Maximum percentage error

To find out just how much the difference between the real and rounded figures is:

  1. Work out maximum. (In the area above we have maximum = 90.7875.)
  2. Take away the rounded answer. (90.7875 - 89.78 = 1.0075.)
  3. Then divide this by the maximum, and times by 100. (1.0075 / 90.7875 x 100 =1.1%.)

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All text copyright © 2006 to EJ Taylor. Page Template created by James Taylor. Site created: 10th April, 2006. Last revised: 2 August, 2015