Solve {c}{sqrt{32}-3sqrt{1/2}}{-4/sqrt{2}}

$$\left. \begin{array} { c } { \sqrt { 32 } - 3 \sqrt { \frac { 1 } { 2 } } } \\ { - \frac { 4 } { \sqrt { 2 } } } \end{array} \right.$$

Evaluate

$$\frac{\sqrt{2}}{2}\approx 0.707106781$$

Short 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}.

$$4\sqrt{2}-3\sqrt{\frac{1}{2}}-\frac{4}{\sqrt{2}}$$

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

$$4\sqrt{2}-3\times \frac{\sqrt{1}}{\sqrt{2}}-\frac{4}{\sqrt{2}}$$

Calculate the square root of 1 and get 1.

$$4\sqrt{2}-3\times \frac{1}{\sqrt{2}}-\frac{4}{\sqrt{2}}$$

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

$$4\sqrt{2}-3\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-\frac{4}{\sqrt{2}}$$

The square of \sqrt{2} is 2.

$$4\sqrt{2}-3\times \frac{\sqrt{2}}{2}-\frac{4}{\sqrt{2}}$$

Express -3\times \frac{\sqrt{2}}{2} as a single fraction.

$$4\sqrt{2}+\frac{-3\sqrt{2}}{2}-\frac{4}{\sqrt{2}}$$

Rationalize the denominator of \frac{4}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$4\sqrt{2}+\frac{-3\sqrt{2}}{2}-\frac{4\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$4\sqrt{2}+\frac{-3\sqrt{2}}{2}-\frac{4\sqrt{2}}{2}$$

Divide 4\sqrt{2} by 2 to get 2\sqrt{2}.

$$4\sqrt{2}+\frac{-3\sqrt{2}}{2}-2\sqrt{2}$$

Combine 4\sqrt{2} and \frac{-3\sqrt{2}}{2} to get \frac{5}{2}\sqrt{2}.

$$\frac{5}{2}\sqrt{2}-2\sqrt{2}$$

Combine \frac{5}{2}\sqrt{2} and -2\sqrt{2} to get \frac{1}{2}\sqrt{2}.

$$\frac{1}{2}\sqrt{2}$$