Solve sqrt{8}+sqrt{1/3}-2sqrt{1/2}

$$\sqrt{ 8 } + \sqrt{ \frac{ 1 }{ 3 } } -2 \sqrt{ \frac{ 1 }{ 2 } }$$

Evaluate

$$\frac{\sqrt{3}\left(\sqrt{6}+1\right)}{3}\approx 1.991563832$$

Short Solution Steps

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}.

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

Rewrite the square root of the division \sqrt{\frac{1}{3}} as the division of square roots \frac{\sqrt{1}}{\sqrt{3}}.

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

Calculate the square root of 1 and get 1.

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

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$2\sqrt{2}+\frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-2\sqrt{\frac{1}{2}}$$

The square of \sqrt{3} is 3.

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

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

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{2} is 2.

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

Cancel out 2 and 2.

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

Combine 2\sqrt{2} and -\sqrt{2} to get \sqrt{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{2} times \frac{3}{3}.

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

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

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