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GCSE Maths
Number
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Surds
Surds are numbers left in 'square root form' (or 'cube root form' etc). They are therefore irrational numbers. The reason we leave them as surds is because in decimal form they would go on forever and so this is a very clumsy way of writing them.
Multiplication of Surds
√5 × √ 15
= √75 (= 15× 5)
= √25 × √ 3
= 5 √ 3
(1 + √3)(2 - √ 8) [The brackets are
expanded as usual]
= 2 - √8 + 2√3 - √24
= 2 - 2√2 + 2√3 - 2√6
Addition and Subtraction of Surds
Adding and subtracting surds are simple- however we need the numbers being square rooted (or cube rooted etc) to be the same.
4√7 - 2√7
= 2√7.
5√2 + 8√2 = 13√2
Note: 5√2 + 3√3 cannot be manipulated because the surds are different (one is
√2 and one is √3).
However, if the number in the square root sign isn't prime, we might be able to split it up in order to simplify an expression.
Example
Simplify √12 + √27
12 = 3 × 4. So √12 = √(3 × 4) = √3 × √4 = 2 × √3.
Similarly, √27 = 3√3.
Hence √12 + √ 27 = 2√3 + 3√3 = 5√3
Rationalising the Denominator
It is untidy to have a fraction which has a surd denominator. This can be 'tidied up' by multiplying the top and bottom of the fraction by a particular expression. This is known as rationalising the denominator, since surds are irrational numbers and so you are changing the denominator from an irrational to a rational number.
Example
Rationalise the denominator of:
a) 1
√2 .
b) 1 + 2
1 - √2
a) Multiply the top and bottom of the fraction by √2. The top will become √2
and the bottom will become 2 (√2 times √2 = 2).
b) In situations like this, look at the bottom of the fraction (the
denominator) and change the sign (in this case change the minus into a plus). Now
multiply the top and bottom of the fraction by this.
Therefore:
| 1 + 2 | = | (1 + 2)(1 + √2) | = | 1 + √2 + 2 + 2√2 | = | 3 + 3√2 |
| 1 - √2 | (1 - √2)(1 + √2) |
1 + √2 - √2 - 2 |
- 1 |
= -3(1 + √2)