Question 38
Let $x$ and $y$ be two positive real numbers. Prove that the expression $$
A=\sqrt{x}+\sqrt{y}+\sqrt{x y}
$$ can be written in the form $$
B=\sqrt{x}+\sqrt{y+x y+2 y \sqrt{x}}
$$ and compare the numbers $$
L=\sqrt{3}+\sqrt{10+2 \sqrt{3}} \quad \text{y} \quad M=\sqrt{5+\sqrt{22}}+\sqrt{8-\sqrt{22}+2 \sqrt{15-3 \sqrt{22}}}
$$
A=\sqrt{x}+\sqrt{y}+\sqrt{x y}
$$ can be written in the form $$
B=\sqrt{x}+\sqrt{y+x y+2 y \sqrt{x}}
$$ and compare the numbers $$
L=\sqrt{3}+\sqrt{10+2 \sqrt{3}} \quad \text{y} \quad M=\sqrt{5+\sqrt{22}}+\sqrt{8-\sqrt{22}+2 \sqrt{15-3 \sqrt{22}}}
$$