Solve (sqrt{8}+3sqrt{1/3})-1/sqrt{2}+frac{sqrt{3}}{2}

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

Evaluate

$$\frac{3\left(\sqrt{2}+\sqrt{3}\right)}{2}\approx 4.719396555$$

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}.

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

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

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{3} is 3.

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

Cancel out 3 and 3.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

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

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

Factor

$$\frac{3 {(\sqrt{2} + \sqrt{3})}}{2} = 4.719396554912959$$