Solve sqrt{32}+sqrt{1/8}-sqrt{8}

$$\sqrt { 32 } + \sqrt { \frac { 1 } { 8 } } - \sqrt { 8 }$$

Evaluate

$$\frac{9\sqrt{2}}{4}\approx 3.181980515$$

Solution Steps

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}.

$$4\sqrt{2}+\sqrt{\frac{1}{8}}-\sqrt{8}$$

Rewrite the square root of the division \sqrt{\frac{1}{8}} as the division of square roots \frac{\sqrt{1}}{\sqrt{8}}.

$$4\sqrt{2}+\frac{\sqrt{1}}{\sqrt{8}}-\sqrt{8}$$

Calculate the square root of 1 and get 1.

$$4\sqrt{2}+\frac{1}{\sqrt{8}}-\sqrt{8}$$

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}.

$$4\sqrt{2}+\frac{1}{2\sqrt{2}}-\sqrt{8}$$

Rationalize the denominator of \frac{1}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$4\sqrt{2}+\frac{\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}-\sqrt{8}$$

The square of \sqrt{2} is 2.

$$4\sqrt{2}+\frac{\sqrt{2}}{2\times 2}-\sqrt{8}$$

Multiply 2 and 2 to get 4.

$$4\sqrt{2}+\frac{\sqrt{2}}{4}-\sqrt{8}$$

Combine 4\sqrt{2} and \frac{\sqrt{2}}{4} to get \frac{17}{4}\sqrt{2}.

$$\frac{17}{4}\sqrt{2}-\sqrt{8}$$

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}.

$$\frac{17}{4}\sqrt{2}-2\sqrt{2}$$

Combine \frac{17}{4}\sqrt{2} and -2\sqrt{2} to get \frac{9}{4}\sqrt{2}.

$$\frac{9}{4}\sqrt{2}$$