Solve sqrt{12}+sqrt{6}*sqrt{1/2}

$$\sqrt { 12 } + \sqrt { 6 } \times \sqrt { \frac { 1 } { 2 } }$$

Evaluate

$$3\sqrt{3}\approx 5.196152423$$

Short Solution Steps

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}.

$$2\sqrt{3}+\sqrt{6}\sqrt{\frac{1}{2}}$$

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

$$2\sqrt{3}+\sqrt{6}\times \frac{\sqrt{1}}{\sqrt{2}}$$

Calculate the square root of 1 and get 1.

$$2\sqrt{3}+\sqrt{6}\times \frac{1}{\sqrt{2}}$$

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

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

The square of \sqrt{2} is 2.

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

Express \sqrt{6}\times \frac{\sqrt{2}}{2} as a single fraction.

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

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

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

Since \frac{2\times 2\sqrt{3}}{2} and \frac{\sqrt{6}\sqrt{2}}{2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2\times 2\sqrt{3}+\sqrt{6}\sqrt{2}.

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

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

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

Divide 6\sqrt{3} by 2 to get 3\sqrt{3}.

$$3\sqrt{3}$$