Simplify \(\sqrt{8}\) to \(2\sqrt{2}\).
\[\frac{{(\sqrt{2}+\sqrt{5})}^{2}+{(2\sqrt{2}-\sqrt{20})}^{2}}{(\sqrt{2}+\sqrt{5})(2\sqrt{2}+\sqrt{20})}\]
Simplify \(\sqrt{20}\) to \(2\sqrt{5}\).
\[\frac{{(\sqrt{2}+\sqrt{5})}^{2}+{(2\sqrt{2}-2\sqrt{5})}^{2}}{(\sqrt{2}+\sqrt{5})(2\sqrt{2}+2\sqrt{5})}\]
Factor out the common term \(2\).
\[\frac{{(\sqrt{2}+\sqrt{5})}^{2}+{(2(\sqrt{2}-\sqrt{5}))}^{2}}{(\sqrt{2}+\sqrt{5})(2\sqrt{2}+2\sqrt{5})}\]
Use Multiplication Distributive Property: \({(xy)}^{a}={x}^{a}{y}^{a}\).
\[\frac{{(\sqrt{2}+\sqrt{5})}^{2}+{2}^{2}{(\sqrt{2}-\sqrt{5})}^{2}}{(\sqrt{2}+\sqrt{5})(2\sqrt{2}+2\sqrt{5})}\]
Expand.
\[\frac{{\sqrt{2}}^{2}+2\sqrt{2}\sqrt{5}+{\sqrt{5}}^{2}+8-8\sqrt{10}+20}{(\sqrt{2}+\sqrt{5})(2\sqrt{2}+2\sqrt{5})}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[\frac{2+2\sqrt{2}\sqrt{5}+{\sqrt{5}}^{2}+8-8\sqrt{10}+20}{(\sqrt{2}+\sqrt{5})(2\sqrt{2}+2\sqrt{5})}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[\frac{2+2\sqrt{2}\sqrt{5}+5+8-8\sqrt{10}+20}{(\sqrt{2}+\sqrt{5})(2\sqrt{2}+2\sqrt{5})}\]
Simplify \(2\sqrt{2}\sqrt{5}\) to \(2\sqrt{2\times 5}\).
\[\frac{2+2\sqrt{2\times 5}+5+8-8\sqrt{10}+20}{(\sqrt{2}+\sqrt{5})(2\sqrt{2}+2\sqrt{5})}\]
Simplify \(2\times 5\) to \(10\).
\[\frac{2+2\sqrt{10}+5+8-8\sqrt{10}+20}{(\sqrt{2}+\sqrt{5})(2\sqrt{2}+2\sqrt{5})}\]
Collect like terms.
\[\frac{(2+5+8+20)+(2\sqrt{10}-8\sqrt{10})}{(\sqrt{2}+\sqrt{5})(2\sqrt{2}+2\sqrt{5})}\]
Simplify \((2+5+8+20)+(2\sqrt{10}-8\sqrt{10})\) to \(35-6\sqrt{10}\).
\[\frac{35-6\sqrt{10}}{(\sqrt{2}+\sqrt{5})(2\sqrt{2}+2\sqrt{5})}\]
Factor out the common term \(2\).
\[\frac{35-6\sqrt{10}}{(\sqrt{2}+\sqrt{5})\times 2(\sqrt{2}+\sqrt{5})}\]
Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[\frac{35-6\sqrt{10}}{{(\sqrt{2}+\sqrt{5})}^{2}\times 2}\]
Regroup terms.
\[\frac{35-6\sqrt{10}}{2{(\sqrt{2}+\sqrt{5})}^{2}}\]
Decimal Form: 0.601383
(35-6*sqrt(10))/(2*(sqrt(2)+sqrt(5))^2)
