Manipulating Expressions with Surds

## Manipulating Expressions with Surds

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)(sqrt(x)). To remove the sqrt(x) as the denominator, multiply the fraction (both numerator and denominator) by sqrt(x), such that:

frac(1)(sqrt(x)) xx frac(sqrt(x))(sqrt(x)) = frac(sqrt(x))(x)

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

sqrt(ab) = sqrt(a) xx sqrt(b) (multiplying two surds);

msqrt(a) + nsqrt(a) = (m + n)sqrt(a) (adding two like surds);

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

## Example 1

Simplify frac(sqrt(x) + 1)(√x).

frac(sqrt(x) + 1)(sqrt(x)) x frac(sqrt(x))(sqrt(x))

= frac(x + sqrt(x))(x)

= 1 + frac(sqrt(x))(x)

Answer: 1 + frac(sqrt(x))(x)

## Example 2

Simplify frac(sqrt(x) + x^2)(sqrt(x))

 frac(sqrt(x) + x^2)(sqrt(x)) xx frac(sqrt(x))(sqrt(x))

= frac(sqrt(x)sqrt(x) + x^2sqrt(x))(sqrt(x)sqrt(x))

= frac(x + x^2sqrt(x))(x)

= 1 + x sqrt(x)

Answer: 1 + x sqrt(x)