# What is a radical conjugate?

May 19, 2016

Assuming that this is a maths question rather than a chemistry question, the radical conjugate of $a + b \sqrt{c}$ is $a - b \sqrt{c}$

#### Explanation:

When simplifying a rational expression such as:

$\frac{1 + \sqrt{3}}{2 + \sqrt{3}}$

we want to rationalise the denominator $\left(2 + \sqrt{3}\right)$ by multiplying by the radical conjugate $\left(2 - \sqrt{3}\right)$, formed by inverting the sign on the radical (square root) term.

So we find:

$\frac{1 + \sqrt{3}}{2 + \sqrt{3}} = \frac{1 + \sqrt{3}}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{\sqrt{3} - 1}{4 - 3} = \sqrt{3} - 1$

This is one use of the difference of squares identity:

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

Specifically:

${a}^{2} - {b}^{2} c = \left(a - b \sqrt{c}\right) \left(a + b \sqrt{c}\right)$

A complex conjugate is actually a special case of the radical conjugate in which the radical is $i = \sqrt{- 1}$