# How do you rationalize the denominator and simplify (4+sqrt3)/(5+sqrt2)?

##### 1 Answer
Mar 13, 2016

To rationalize the denominator we must get rid of any radicals in the denominator.

#### Explanation:

We can rationalize the denominator of a binomial by multiplying the numerator and the denominator by the conjugate of the denominator.

Definition: the conjugate is what forms a difference of squares. Ex the conjugate of $\left(a + b\right)$ is $\left(a - b\right)$, since $\left(a + b\right) \left(a - b\right) = {a}^{2} - {b}^{2}$. Essentially, to find the conjugate all you have to do is simply to switch the sign $+ \mathmr{and} -$ sign between the two terms in the binomial.

Thus, the conjugate of $5 + \sqrt{2}$ is $5 - \sqrt{2}$.

$\frac{4 + \sqrt{3}}{5 + \sqrt{2}} \times \frac{5 - \sqrt{2}}{5 - \sqrt{2}}$

Don't forget that you cannot multiply non radicals with radicals. E.g $3 \times \sqrt{8} \ne \sqrt{24}$ but is simply $3 \sqrt{8}$.

-> (20 + 5sqrt(3) - 4sqrt(2) - sqrt(6))/(25 + 2sqrt(5) - 2sqrt(5) - sqrt(4)

= $\frac{20 + 5 \sqrt{3} - 4 \sqrt{2} - \sqrt{6}}{23}$

Practice exercises:

Rationalize the denominator of $\frac{3 \sqrt{5} - 2 \sqrt{7}}{2 \sqrt{3} - 4 \sqrt{2}}$

Good luck!