Solve sqrt{32}-3sqrt{1/2}+2

$$\sqrt { 32 } - 3 \sqrt { \frac { 1 } { 2 } } + 2$$

Evaluate

$$\frac{5\sqrt{2}}{2}+2\approx 5.535533906$$

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}.

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

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

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 4\sqrt{2}+2 times \frac{2}{2}.

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

Since \frac{2\left(4\sqrt{2}+2\right)}{2} and \frac{-3\sqrt{2}}{2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2\left(4\sqrt{2}+2\right)-3\sqrt{2}.

$$\frac{8\sqrt{2}+4-3\sqrt{2}}{2}$$

Do the calculations in 8\sqrt{2}+4-3\sqrt{2}.

$$\frac{5\sqrt{2}+4}{2}$$