 #### Revision Notes

##### What is a surd?

A surd is the square root of a non-square integer.

##### What can we do with surds?

1. Multiplying surds – you can multiply numbers under square roots.
eg √3 × √5 = √3×5 = √15
2. Dividing surds – you can divide numbers under square roots.
eg √21 ÷ √7= √21 ÷ 7 = √3
3. Factorising surds – you can factorise numbers under square roots.
eg √35 = √5 × 7 = √5 × √7
4. Simplifying surds – separate out a square factor and square root it!
eg √48 = √16 × 3 = √16 × √3 = 4 × √3 = 4√3
5. Adding or subtracting surds is very like adding or subtracting letters in algebra – you can only add or subtract multiples of “like” surds.
eg 3√5 + 8√5 = 11√5 or 7√3 – 4√3 = 3√3

Be very careful here!  You can not add or subtract numbers under square roots.  Think about √9 + √4= 3 + 2 = 5.  It is not equal to √9 + 4= √13  = 3.60555…

6. All other algebraic rules apply – surds can be treated like letters (as in 4. above) and like numbers (as in 1. and 2. above).

If you need to do anything more complicated (like rationalising the denominator), see the next set of Notes.   Link here to 1.12.2 doc?

#### Example

e.g. Write √54 – √24 in the form p√q where p and q are integers

and q has no square factors.

1. √54 – √24 = √(9 × 6) – √(4 × 6)

= √9 × √6 – √4 × √6

= 3 × √6 – 2 × √6

1.     = 3√6 – 2√6

= √6

so p = 1 and q = 6

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