Manipulating Expressions with Surds

Manipulating Expressions with Surds

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 √ba, 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).

Examples

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`