Solve sqrt{2/3}divsqrt{1/18}*sqrt{24}

$$\sqrt{ \frac{ 2 }{ 3 } } \div \sqrt{ \frac{ 1 }{ 18 } } \times \sqrt{ 24 }$$

Evaluate

$$4\sqrt{18}\approx 16.970562748$$

Solution Steps

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

$$\frac{\frac{\sqrt{2}}{\sqrt{3}}}{\sqrt{\frac{1}{18}}}\sqrt{24}$$

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

$$\frac{\frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{\sqrt{\frac{1}{18}}}\sqrt{24}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{\sqrt{2}\sqrt{3}}{3}}{\sqrt{\frac{1}{18}}}\sqrt{24}$$

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

$$\frac{\frac{\sqrt{6}}{3}}{\sqrt{\frac{1}{18}}}\sqrt{24}$$

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

$$\frac{\frac{\sqrt{6}}{3}}{\frac{\sqrt{1}}{\sqrt{18}}}\sqrt{24}$$

Calculate the square root of 1 and get 1.

$$\frac{\frac{\sqrt{6}}{3}}{\frac{1}{\sqrt{18}}}\sqrt{24}$$

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}.

$$\frac{\frac{\sqrt{6}}{3}}{\frac{1}{3\sqrt{2}}}\sqrt{24}$$

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

$$\frac{\frac{\sqrt{6}}{3}}{\frac{\sqrt{2}}{3\left(\sqrt{2}\right)^{2}}}\sqrt{24}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{\sqrt{6}}{3}}{\frac{\sqrt{2}}{3\times 2}}\sqrt{24}$$

Multiply 3 and 2 to get 6.

$$\frac{\frac{\sqrt{6}}{3}}{\frac{\sqrt{2}}{6}}\sqrt{24}$$

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

$$\frac{\sqrt{6}\times 6}{3\sqrt{2}}\sqrt{24}$$

Cancel out 3 in both numerator and denominator.

$$\frac{2\sqrt{6}}{\sqrt{2}}\sqrt{24}$$

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

$$\frac{2\sqrt{6}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\sqrt{24}$$

The square of \sqrt{2} is 2.

$$\frac{2\sqrt{6}\sqrt{2}}{2}\sqrt{24}$$

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}.

$$\frac{2\sqrt{2}\sqrt{3}\sqrt{2}}{2}\sqrt{24}$$

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

$$\frac{2\times 2\sqrt{3}}{2}\sqrt{24}$$

Cancel out 2 and 2.

$$2\sqrt{3}\sqrt{24}$$

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

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

Multiply 2 and 2 to get 4.

$$4\sqrt{3}\sqrt{6}$$

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

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

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

$$4\times 3\sqrt{2}$$

Multiply 4 and 3 to get 12.

$$12\sqrt{2}$$