Solve sqrt{1/2}+3sqrt{1/3}-3/2(sqrt{8}-sqrt{108})=

$$\sqrt { \frac { 1 } { 2 } } + 3 \sqrt { \frac { 1 } { 3 } } - \frac { 3 } { 2 } ( \sqrt { 8 } - \sqrt { 108 } ) =$$

Evaluate

$$-\frac{5\sqrt{2}}{2}+10\sqrt{3}\approx 13.78497417$$

Short Solution Steps

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

$$\frac{\sqrt{1}}{\sqrt{2}}+3\sqrt{\frac{1}{3}}-\frac{3}{2}\left(\sqrt{8}-\sqrt{108}\right)$$

Calculate the square root of 1 and get 1.

$$\frac{1}{\sqrt{2}}+3\sqrt{\frac{1}{3}}-\frac{3}{2}\left(\sqrt{8}-\sqrt{108}\right)$$

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

$$\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+3\sqrt{\frac{1}{3}}-\frac{3}{2}\left(\sqrt{8}-\sqrt{108}\right)$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}}{2}+3\sqrt{\frac{1}{3}}-\frac{3}{2}\left(\sqrt{8}-\sqrt{108}\right)$$

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

$$\frac{\sqrt{2}}{2}+3\times \frac{\sqrt{1}}{\sqrt{3}}-\frac{3}{2}\left(\sqrt{8}-\sqrt{108}\right)$$

Calculate the square root of 1 and get 1.

$$\frac{\sqrt{2}}{2}+3\times \frac{1}{\sqrt{3}}-\frac{3}{2}\left(\sqrt{8}-\sqrt{108}\right)$$

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

$$\frac{\sqrt{2}}{2}+3\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-\frac{3}{2}\left(\sqrt{8}-\sqrt{108}\right)$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{2}}{2}+3\times \frac{\sqrt{3}}{3}-\frac{3}{2}\left(\sqrt{8}-\sqrt{108}\right)$$

Cancel out 3 and 3.

$$\frac{\sqrt{2}}{2}+\sqrt{3}-\frac{3}{2}\left(\sqrt{8}-\sqrt{108}\right)$$

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

$$\frac{\sqrt{2}}{2}+\frac{2\sqrt{3}}{2}-\frac{3}{2}\left(\sqrt{8}-\sqrt{108}\right)$$

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

$$\frac{\sqrt{2}+2\sqrt{3}}{2}-\frac{3}{2}\left(\sqrt{8}-\sqrt{108}\right)$$

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}.

$$\frac{\sqrt{2}+2\sqrt{3}}{2}-\frac{3}{2}\left(2\sqrt{2}-\sqrt{108}\right)$$

Factor 108=6^{2}\times 3. Rewrite the square root of the product \sqrt{6^{2}\times 3} as the product of square roots \sqrt{6^{2}}\sqrt{3}. Take the square root of 6^{2}.

$$\frac{\sqrt{2}+2\sqrt{3}}{2}-\frac{3}{2}\left(2\sqrt{2}-6\sqrt{3}\right)$$

Use the distributive property to multiply -\frac{3}{2} by 2\sqrt{2}-6\sqrt{3}.

$$\frac{\sqrt{2}+2\sqrt{3}}{2}-\frac{3}{2}\times 2\sqrt{2}-\frac{3}{2}\left(-6\right)\sqrt{3}$$

Cancel out 2 and 2.

$$\frac{\sqrt{2}+2\sqrt{3}}{2}-3\sqrt{2}-\frac{3}{2}\left(-6\right)\sqrt{3}$$

Express -\frac{3}{2}\left(-6\right) as a single fraction.

$$\frac{\sqrt{2}+2\sqrt{3}}{2}-3\sqrt{2}+\frac{-3\left(-6\right)}{2}\sqrt{3}$$

Multiply -3 and -6 to get 18.

$$\frac{\sqrt{2}+2\sqrt{3}}{2}-3\sqrt{2}+\frac{18}{2}\sqrt{3}$$

Divide 18 by 2 to get 9.

$$\frac{\sqrt{2}+2\sqrt{3}}{2}-3\sqrt{2}+9\sqrt{3}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -3\sqrt{2}+9\sqrt{3} times \frac{2}{2}.

$$\frac{\sqrt{2}+2\sqrt{3}}{2}+\frac{2\left(-3\sqrt{2}+9\sqrt{3}\right)}{2}$$

Since \frac{\sqrt{2}+2\sqrt{3}}{2} and \frac{2\left(-3\sqrt{2}+9\sqrt{3}\right)}{2} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{2}+2\sqrt{3}+2\left(-3\sqrt{2}+9\sqrt{3}\right)}{2}$$

Do the multiplications in \sqrt{2}+2\sqrt{3}+2\left(-3\sqrt{2}+9\sqrt{3}\right).

$$\frac{\sqrt{2}+2\sqrt{3}-6\sqrt{2}+18\sqrt{3}}{2}$$

Do the calculations in \sqrt{2}+2\sqrt{3}-6\sqrt{2}+18\sqrt{3}.

$$\frac{-5\sqrt{2}+20\sqrt{3}}{2}$$

Factor

$$\frac{5 {(4 \sqrt{3} - \sqrt{2})}}{2} = 13.784974169756035$$