Solve {left(sqrt{9+sqrt{80}}right)}^2+2+{left(sqrt{9-sqrt{80}}right)}^2

$${ \left( \sqrt{ 9+ \sqrt{ 80 } } \right) }^{ 2 } +2+ { \left( \sqrt{ 9- \sqrt{ 80 } } \right) }^{ 2 }$$

Evaluate

$$20$$

Solution Steps

Factor 80=4^{2}\times 5. Rewrite the square root of the product \sqrt{4^{2}\times 5} as the product of square roots \sqrt{4^{2}}\sqrt{5}. Take the square root of 4^{2}.

$$\left(\sqrt{9+4\sqrt{5}}\right)^{2}+2+\left(\sqrt{9-\sqrt{80}}\right)^{2}$$

Calculate \sqrt{9+4\sqrt{5}} to the power of 2 and get 9+4\sqrt{5}.

$$9+4\sqrt{5}+2+\left(\sqrt{9-\sqrt{80}}\right)^{2}$$

Add 9 and 2 to get 11.

$$11+4\sqrt{5}+\left(\sqrt{9-\sqrt{80}}\right)^{2}$$

Factor 80=4^{2}\times 5. Rewrite the square root of the product \sqrt{4^{2}\times 5} as the product of square roots \sqrt{4^{2}}\sqrt{5}. Take the square root of 4^{2}.

$$11+4\sqrt{5}+\left(\sqrt{9-4\sqrt{5}}\right)^{2}$$

Calculate \sqrt{9-4\sqrt{5}} to the power of 2 and get 9-4\sqrt{5}.

$$11+4\sqrt{5}+9-4\sqrt{5}$$

Add 11 and 9 to get 20.

$$20+4\sqrt{5}-4\sqrt{5}$$

Combine 4\sqrt{5} and -4\sqrt{5} to get 0.

$$20$$

Factor

$$2^{2}\times 5$$