Solve sqrt{8}+sqrt{12}-2/sqrt{2}+3sqrt{1/3}

$$\sqrt { 8 } + \sqrt { 12 } - \frac { 2 } { \sqrt { 2 } } + 3 \sqrt { \frac { 1 } { 3 } }$$

Evaluate

$$\sqrt{2}+3\sqrt{3}\approx 6.610365985$$

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}.

$$2\sqrt{2}+\sqrt{12}-\frac{2}{\sqrt{2}}+3\sqrt{\frac{1}{3}}$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

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

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

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

The square of \sqrt{2} is 2.

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

Cancel out 2 and 2.

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

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

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{3} is 3.

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

Cancel out 3 and 3.

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

Combine 2\sqrt{3} and \sqrt{3} to get 3\sqrt{3}.

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

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

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