Solve 6sqrt{2/3}div3sqrt{1/18}

$$6 \sqrt { \frac { 2 } { 3 } } \div 3 \sqrt { \frac { 1 } { 18 } }$$

Evaluate

$$\frac{2\sqrt{3}}{9}\approx 0.384900179$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{2}{3}} as the division of square roots \frac{\sqrt{2}}{\sqrt{3}}.

$$\frac{6\times \frac{\sqrt{2}}{\sqrt{3}}}{3}\sqrt{\frac{1}{18}}$$

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

$$\frac{6\times \frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{3}\sqrt{\frac{1}{18}}$$

The square of \sqrt{3} is 3.

$$\frac{6\times \frac{\sqrt{2}\sqrt{3}}{3}}{3}\sqrt{\frac{1}{18}}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{6\times \frac{\sqrt{6}}{3}}{3}\sqrt{\frac{1}{18}}$$

Cancel out 3, the greatest common factor in 6 and 3.

$$\frac{2\sqrt{6}}{3}\sqrt{\frac{1}{18}}$$

Rewrite the square root of the division \sqrt{\frac{1}{18}} as the division of square roots \frac{\sqrt{1}}{\sqrt{18}}.

$$\frac{2\sqrt{6}}{3}\times \frac{\sqrt{1}}{\sqrt{18}}$$

Calculate the square root of 1 and get 1.

$$\frac{2\sqrt{6}}{3}\times \frac{1}{\sqrt{18}}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply 3 and 2 to get 6.

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

Multiply \frac{2\sqrt{6}}{3} times \frac{\sqrt{2}}{6} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 2 in both numerator and denominator.

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

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

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

Multiply 3 and 3 to get 9.

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