GCSE(F), GCSE(H),

**Surds** can occur in algebraic expressions. When a surd involves a fraction, it should not be left with a square root as a denominator. As an example, consider `frac(1)(√x)`. To remove the √*x* as the denominator, multiply the fraction (both numerator and denominator) by √*x*, such that:

`frac(1)(√x) xx frac(√x)(√x) = frac(√x)(x)`

Surds can be resolved without multiplying the fraction, such as `frac(b√a)(√b)`. In this instance, completing the division by √*b* gives an answer of √*b*√*a*, which can be written as `sqrt(ab)``.

The rules for manipulating surds algebraically are the same as those used for manipulating surds arithmetically:

• √*ab* = √*a* x √*b* (multiplying two surds);

• m√*a* + n√*a* = (m + n)√*a* (adding two like surds);

• `sqrt(frac(a)(b))` = `frac(√a)(√b)` (dividing two surds).

1. Simplify `frac(√x + 1)(√x)`.

Answer: 1 + `frac(√x)(x)`

`frac(√x + 1)(√x)` x `frac(√x)(√x)`

= `frac(x + √x)(x)`

= `1 + frac(√x)(x)`

2. Simplify `frac(√x + x^2)(√x)`

Answer: 1 + x√x

` frac(√x + x^2)(√x) xx frac(√x)(√x)`

`= frac(√x√x + x^2√x)(√x√x)`

`= frac(x + x^2√x)(x)`

`= 1 + x√x`

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