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`