Question

Write whether $\frac{2\sqrt{45}+3\sqrt{20}}{2\sqrt{5}}$ on simplification gives as rational or an irrational number

Answer 1

Consider the problem

$\begin{array}{c}\frac{2\sqrt{45}+3\sqrt{20}}{2\sqrt{5}}\\ =\frac{2\sqrt{3\times 3\times 5}+3\sqrt{5\times 2\times 2}}{2\sqrt{5}}\\ =\frac{2\times 3\sqrt{5}+3\times 2\sqrt{5}}{2\sqrt{5}}\\ =\frac{6\sqrt{5}+6\sqrt{5}}{2\sqrt{5}}\\ =\frac{12\sqrt{5}}{2\sqrt{5}}\\ =6\end{array}$

Hence, $6$, which is a rational number