# What is the radical conjugate of 12-sqrt(5) ?

Jun 25, 2016

$12 + \sqrt{5}$

#### Explanation:

The conjugate of $12 - \sqrt{5}$ is $12 + \sqrt{5}$.

This has the property that: $\left(12 - \sqrt{5}\right) \left(12 + \sqrt{5}\right)$ is rational:

$\left(12 - \sqrt{5}\right) \left(12 + \sqrt{5}\right) = {12}^{2} - {\left(\sqrt{5}\right)}^{2} = 144 - 5 = 139$

Typical examples where you would use the conjugate would be:

• When rationalising the denominator of a quotient.
• When looking at zeros of a polynomial with rational (typically integer) coefficients.

For example:

$\frac{2 + 3 \sqrt{5}}{12 - \sqrt{5}}$

$= \frac{\left(2 + 3 \sqrt{5}\right) \left(12 + \sqrt{5}\right)}{\left(12 - \sqrt{5}\right) \left(12 + \sqrt{5}\right)}$

$= \frac{24 + 2 \sqrt{5} + 36 \sqrt{5} + 15}{144 - 5}$

$= \frac{39 + 38 \sqrt{5}}{139}$

The simplest polynomial with rational coefficients and zero $12 - \sqrt{5}$ is:

$\left(x - \left(12 - \sqrt{5}\right)\right) \left(x - \left(12 + \sqrt{5}\right)\right) = {x}^{2} - 24 x + 139$