The process 12, 22, 32, 42, ... give us the square numbers 1, 4, 9, 16, ....
and the process 13, 23, 33, 43, ... give us the cube numbers 1, 8, 27, 64, ...
So reversing the process,
and
But what about
… ?
Eg. 
correct to 31 decimal
places.
And all of the integers (whole numbers) between the square numbers have square roots that are irrational, that is numbers which have an unending decimal part that has no pattern.
The same applies to the cube roots of the integers between the cube numbers.
The irrational numbers which can be written using a root sign are called surds.
So for the square root of 2, it is quicker and more accurate to write the
suds form,
, than
1.4142135623730950488016887242097
Because the square root of 2 is
, then
If the number under the square root symbol can be factorised, then the factors can be square rooted separately.
Egs 
So: 
Also, in a fraction, the top and the bottom of the fraction can be square rooted separately.
Eg. 
Putting these techniques together allows some expressions to be simplified.
Eg.
It is not usual to leave an answer with surds on the bottom of a fraction. So the fraction would be rationalised by multiplying top and bottom of the fraction (which is just the reverse of cancelling down) by any surds on the bottom of the fraction.
Eg. 
Practise with these questions.
| A
Simplify the following. |
B
Rationalise the following 1.
2. |
The method to rationalise the denominators can be extended to examples where
the denominator has two, rather than one term, eg.
Here it is necessary to multiply top and bottom of the fraction by whatever is on the bottom, but with the opposite sign.
Although the numerator is now irrational, the denominator is rational and so the expression is rationalised.
| C
Rationalise |
|
