Answer
$\dfrac{4\sqrt{5}+\sqrt{2}}{2\sqrt{5}-\sqrt{2}}=\dfrac{7+\sqrt{10}}{3}$
Work Step by Step
$\dfrac{4\sqrt{5}+\sqrt{2}}{2\sqrt{5}-\sqrt{2}}$
Multiply the numerator and the denominator of this expression by the conjugate of the denominator:
$\dfrac{4\sqrt{5}+\sqrt{2}}{2\sqrt{5}-\sqrt{2}}=\dfrac{4\sqrt{5}+\sqrt{2}}{2\sqrt{5}-\sqrt{2}}\cdot\dfrac{2\sqrt{5}+\sqrt{2}}{2\sqrt{5}+\sqrt{2}}=...$
$...=\dfrac{(4\sqrt{5}+\sqrt{2})(2\sqrt{5}+\sqrt{2})}{(2\sqrt{5})^{2}-(\sqrt{2})^{2}}=...$
$...=\dfrac{8\sqrt{5^{2}}+4\sqrt{10}+2\sqrt{10}+\sqrt{2^{2}}}{4(5)-2}=...$
$...=\dfrac{8(5)+6\sqrt{10}+2}{20-2}=\dfrac{40+2+6\sqrt{10}}{18}=\dfrac{42+6\sqrt{10}}{18}=...$
Take out common factor $6$ from the numerator and simplify:
$...=\dfrac{6(7+\sqrt{10})}{18}=\dfrac{7+\sqrt{10}}{3}$
