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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)`

See also Calculating Exactly with Surds and Surd Denominators