Solve 1/2sqrt{3}divsqrt{1/12}*sqrt{27}

$$\frac { 1 } { 2 } \sqrt { 3 } \div \sqrt { \frac { 1 } { 12 } } \times \sqrt { 27 }$$

Evaluate

$$9\sqrt{3}\approx 15.588457268$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{12}} as the division of square roots \frac{\sqrt{1}}{\sqrt{12}}.

$$\frac{\frac{1}{2}\sqrt{3}}{\frac{\sqrt{1}}{\sqrt{12}}}\sqrt{27}$$

Calculate the square root of 1 and get 1.

$$\frac{\frac{1}{2}\sqrt{3}}{\frac{1}{\sqrt{12}}}\sqrt{27}$$

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}.

$$\frac{\frac{1}{2}\sqrt{3}}{\frac{1}{2\sqrt{3}}}\sqrt{27}$$

Rationalize the denominator of \frac{1}{2\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\frac{1}{2}\sqrt{3}}{\frac{\sqrt{3}}{2\left(\sqrt{3}\right)^{2}}}\sqrt{27}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{1}{2}\sqrt{3}}{\frac{\sqrt{3}}{2\times 3}}\sqrt{27}$$

Multiply 2 and 3 to get 6.

$$\frac{\frac{1}{2}\sqrt{3}}{\frac{\sqrt{3}}{6}}\sqrt{27}$$

Divide \frac{1}{2}\sqrt{3} by \frac{\sqrt{3}}{6} by multiplying \frac{1}{2}\sqrt{3} by the reciprocal of \frac{\sqrt{3}}{6}.

$$\frac{\frac{1}{2}\sqrt{3}\times 6}{\sqrt{3}}\sqrt{27}$$

Rationalize the denominator of \frac{\frac{1}{2}\sqrt{3}\times 6}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\frac{1}{2}\sqrt{3}\times 6\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\sqrt{27}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{1}{2}\sqrt{3}\times 6\sqrt{3}}{3}\sqrt{27}$$

Multiply \frac{1}{2} and 6 to get \frac{6}{2}.

$$\frac{\frac{6}{2}\sqrt{3}\sqrt{3}}{3}\sqrt{27}$$

Divide 6 by 2 to get 3.

$$\frac{3\sqrt{3}\sqrt{3}}{3}\sqrt{27}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{3\times 3}{3}\sqrt{27}$$

Multiply 3 and 3 to get 9.

$$\frac{9}{3}\sqrt{27}$$

Divide 9 by 3 to get 3.

$$3\sqrt{27}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$3\times 3\sqrt{3}$$

Multiply 3 and 3 to get 9.

$$9\sqrt{3}$$