Solve 3sqrt{8}+{left(sqrt{2}+sqrt{32}right)}^2

$$3 \sqrt{ 8 } + { \left( \sqrt{ 2 } + \sqrt{ 32 } \right) }^{ 2 }$$

Evaluate

$$6\sqrt{2}+50\approx 58.485281374$$

Solution Steps

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$3\times 2\sqrt{2}+\left(\sqrt{2}+\sqrt{32}\right)^{2}$$

Multiply 3 and 2 to get 6.

$$6\sqrt{2}+\left(\sqrt{2}+\sqrt{32}\right)^{2}$$

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

$$6\sqrt{2}+\left(\sqrt{2}+4\sqrt{2}\right)^{2}$$

Combine \sqrt{2} and 4\sqrt{2} to get 5\sqrt{2}.

$$6\sqrt{2}+\left(5\sqrt{2}\right)^{2}$$

Expand \left(5\sqrt{2}\right)^{2}.

$$6\sqrt{2}+5^{2}\left(\sqrt{2}\right)^{2}$$

Calculate 5 to the power of 2 and get 25.

$$6\sqrt{2}+25\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$6\sqrt{2}+25\times 2$$

Multiply 25 and 2 to get 50.

$$6\sqrt{2}+50$$