Question

Rationalise $\frac{3}{2\sqrt{5}-3\sqrt{2}}$

Answer 1

Given $\frac{3}{2\sqrt{5}-3\sqrt{2}}$

Multiply the numerator and denominator by $(2\sqrt{5}+3\sqrt{2})$

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

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

Using the identity $(a-b)(a+b)=a^2-b^2$

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

$=\frac{3(2\sqrt{5}+3\sqrt{2})}{20-18}$

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