Solve sqrt{2}+sqrt{8}+sqrt{18}+sqrt{32}

$$\sqrt { 2 } + \sqrt { 8 } + \sqrt { 18 } + \sqrt { 32 }$$

Evaluate

$$10\sqrt{2}\approx 14.142135624$$

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}.

$$\sqrt{2}+2\sqrt{2}+\sqrt{18}+\sqrt{32}$$

Combine \sqrt{2} and 2\sqrt{2} to get 3\sqrt{2}.

$$3\sqrt{2}+\sqrt{18}+\sqrt{32}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$3\sqrt{2}+3\sqrt{2}+\sqrt{32}$$

Combine 3\sqrt{2} and 3\sqrt{2} to get 6\sqrt{2}.

$$6\sqrt{2}+\sqrt{32}$$

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}.

$$6\sqrt{2}+4\sqrt{2}$$

Combine 6\sqrt{2} and 4\sqrt{2} to get 10\sqrt{2}.

$$10\sqrt{2}$$