GCSE(H),

When a surd forms part of a denominator in a fraction, it must be removed.

The surd can be removed by multiplying both the numerator and the denominator by the value of the surd. For example, multiply `frac(√a)(√b)` by `frac(√b)(√b)`, and simplify to `frac(a√b)(b)`.

This process is called **rationalising the denominator**, as the denominator is set to be a rational number.

1. Rationalise and simplify: `frac(27)(√3)`

Answer: 9√3

`frac(27)(√3)`

`= frac(27√3)(√3√3)`

`= frac(27√3)(3)`

`= 9√3`

2. Rationalise `frac(8√3)(√6)`.

Answer: 4√2

`frac(8√3)(√6)`

`= frac(8√3)(√2√3)`

`= frac(8)(√2)`

`= frac(8√2)(√2√2)`

`= frac(8√2)(2)`

`= 4√2`

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