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`