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

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

Evaluate

$$\frac{7\sqrt{2}}{2}\approx 4.949747468$$

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}.

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

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

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

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

$$5\sqrt{2}+\frac{-3\sqrt{2}}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5\sqrt{2} times \frac{2}{2}.

$$\frac{2\times 5\sqrt{2}}{2}+\frac{-3\sqrt{2}}{2}$$

Since \frac{2\times 5\sqrt{2}}{2} and \frac{-3\sqrt{2}}{2} have the same denominator, add them by adding their numerators.

$$\frac{2\times 5\sqrt{2}-3\sqrt{2}}{2}$$

Do the multiplications in 2\times 5\sqrt{2}-3\sqrt{2}.

$$\frac{10\sqrt{2}-3\sqrt{2}}{2}$$

Do the calculations in 10\sqrt{2}-3\sqrt{2}.

$$\frac{7\sqrt{2}}{2}$$