Solve sqrt{32}+sqrt{1/2}-6sqrt{1/8}

$$\sqrt { 32 } + \sqrt { \frac { 1 } { 2 } } - 6 \sqrt { \frac { 1 } { 8 } }$$

Evaluate

$$3\sqrt{2}\approx 4.242640687$$

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}.

$$4\sqrt{2}+\sqrt{\frac{1}{2}}-6\sqrt{\frac{1}{8}}$$

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

$$4\sqrt{2}+\frac{\sqrt{1}}{\sqrt{2}}-6\sqrt{\frac{1}{8}}$$

Calculate the square root of 1 and get 1.

$$4\sqrt{2}+\frac{1}{\sqrt{2}}-6\sqrt{\frac{1}{8}}$$

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

$$4\sqrt{2}+\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-6\sqrt{\frac{1}{8}}$$

The square of \sqrt{2} is 2.

$$4\sqrt{2}+\frac{\sqrt{2}}{2}-6\sqrt{\frac{1}{8}}$$

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

$$\frac{9}{2}\sqrt{2}-6\sqrt{\frac{1}{8}}$$

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

$$\frac{9}{2}\sqrt{2}-6\times \frac{\sqrt{1}}{\sqrt{8}}$$

Calculate the square root of 1 and get 1.

$$\frac{9}{2}\sqrt{2}-6\times \frac{1}{\sqrt{8}}$$

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}.

$$\frac{9}{2}\sqrt{2}-6\times \frac{1}{2\sqrt{2}}$$

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

$$\frac{9}{2}\sqrt{2}-6\times \frac{\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{9}{2}\sqrt{2}-6\times \frac{\sqrt{2}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{9}{2}\sqrt{2}-6\times \frac{\sqrt{2}}{4}$$

Express -6\times \frac{\sqrt{2}}{4} as a single fraction.

$$\frac{9}{2}\sqrt{2}+\frac{-6\sqrt{2}}{4}$$

Divide -6\sqrt{2} by 4 to get -\frac{3}{2}\sqrt{2}.

$$\frac{9}{2}\sqrt{2}-\frac{3}{2}\sqrt{2}$$

Combine \frac{9}{2}\sqrt{2} and -\frac{3}{2}\sqrt{2} to get 3\sqrt{2}.

$$3\sqrt{2}$$