Solve (sqrt{1/2}+sqrt{3})-(sqrt{2}-sqrt{12})

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

Evaluate

$$-\frac{\sqrt{2}}{2}+3\sqrt{3}\approx 4.489045642$$

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}}.

$$\frac{\sqrt{1}}{\sqrt{2}}+\sqrt{3}-\left(\sqrt{2}-\sqrt{12}\right)$$

Calculate the square root of 1 and get 1.

$$\frac{1}{\sqrt{2}}+\sqrt{3}-\left(\sqrt{2}-\sqrt{12}\right)$$

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

$$\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+\sqrt{3}-\left(\sqrt{2}-\sqrt{12}\right)$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}}{2}+\sqrt{3}-\left(\sqrt{2}-\sqrt{12}\right)$$

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

$$\frac{\sqrt{2}}{2}+\frac{2\sqrt{3}}{2}-\left(\sqrt{2}-\sqrt{12}\right)$$

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

$$\frac{\sqrt{2}+2\sqrt{3}}{2}-\left(\sqrt{2}-\sqrt{12}\right)$$

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}.

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

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

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

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

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

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

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

Do the calculations in \sqrt{2}+2\sqrt{3}-2\sqrt{2}+4\sqrt{3}.

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