Solve sqrt{54}*sqrt{1/2}+sqrt{12}

$$\sqrt{ 54 } \times \sqrt{ \frac{ 1 }{ 2 } } + \sqrt{ 12 }$$

Evaluate

$$5\sqrt{3}\approx 8.660254038$$

Short Solution Steps

Factor 54=3^{2}\times 6. Rewrite the square root of the product \sqrt{3^{2}\times 6} as the product of square roots \sqrt{3^{2}}\sqrt{6}. Take the square root of 3^{2}.

$$3\sqrt{6}\sqrt{\frac{1}{2}}+\sqrt{12}$$

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

$$3\sqrt{6}\times \frac{\sqrt{1}}{\sqrt{2}}+\sqrt{12}$$

Calculate the square root of 1 and get 1.

$$3\sqrt{6}\times \frac{1}{\sqrt{2}}+\sqrt{12}$$

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

$$3\sqrt{6}\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+\sqrt{12}$$

The square of \sqrt{2} is 2.

$$3\sqrt{6}\times \frac{\sqrt{2}}{2}+\sqrt{12}$$

Express 3\times \frac{\sqrt{2}}{2} as a single fraction.

$$\frac{3\sqrt{2}}{2}\sqrt{6}+\sqrt{12}$$

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}.

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

Express \frac{3\sqrt{2}}{2}\sqrt{6} as a single fraction.

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

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

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

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

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

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

$$\frac{6\sqrt{3}+4\sqrt{3}}{2}$$

Do the calculations in 6\sqrt{3}+4\sqrt{3}.

$$\frac{10\sqrt{3}}{2}$$