# How do you rationalize (2sqrt5)/ ( 2sqrt5 + 3sqrt2)?

May 11, 2018

$\frac{2 \sqrt{5}}{2 \sqrt{5} + 3 \sqrt{2}} = 10 - 3 \sqrt{10}$

#### Explanation:

Note that the difference of squares identity tells us that:

${A}^{2} - {B}^{2} = \left(A - B\right) \left(A + B\right)$

Hence we can rationalize the denominator of the given expression by multiplying both numerator and denominator by $2 \sqrt{5} - 3 \sqrt{2}$ ...

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

$\textcolor{w h i t e}{\frac{2 \sqrt{5}}{2 \sqrt{5} + 3 \sqrt{2}}} = \frac{{\left(2 \sqrt{5}\right)}^{2} - \left(2 \sqrt{5}\right) \left(3 \sqrt{2}\right)}{{\left(2 \sqrt{5}\right)}^{2} - {\left(3 \sqrt{2}\right)}^{2}}$

$\textcolor{w h i t e}{\frac{2 \sqrt{5}}{2 \sqrt{5} + 3 \sqrt{2}}} = \frac{20 - 6 \sqrt{10}}{20 - 18}$

$\textcolor{w h i t e}{\frac{2 \sqrt{5}}{2 \sqrt{5} + 3 \sqrt{2}}} = 10 - 3 \sqrt{10}$