Solve frac{frac{3sqrt{3}}{sqrt{2}}8/sqrt{18}}{frac{sqrt{6}}{sqrt{5}}}

$$\frac{ \frac{ 3 \sqrt{ 3 } }{ \sqrt{ 2 } } \frac{ 8 }{ \sqrt{ 18 } } }{ \frac{ \sqrt{ 6 } }{ \sqrt{ 5 } } }$$

Evaluate

$$2\sqrt{10}\approx 6.32455532$$

Solution Steps

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

$$\frac{\frac{3\sqrt{3}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\times \frac{8}{\sqrt{18}}}{\frac{\sqrt{6}}{\sqrt{5}}}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{3\sqrt{3}\sqrt{2}}{2}\times \frac{8}{\sqrt{18}}}{\frac{\sqrt{6}}{\sqrt{5}}}$$

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

$$\frac{\frac{3\sqrt{6}}{2}\times \frac{8}{\sqrt{18}}}{\frac{\sqrt{6}}{\sqrt{5}}}$$

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}.

$$\frac{\frac{3\sqrt{6}}{2}\times \frac{8}{3\sqrt{2}}}{\frac{\sqrt{6}}{\sqrt{5}}}$$

Rationalize the denominator of \frac{8}{3\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\frac{3\sqrt{6}}{2}\times \frac{8\sqrt{2}}{3\left(\sqrt{2}\right)^{2}}}{\frac{\sqrt{6}}{\sqrt{5}}}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{3\sqrt{6}}{2}\times \frac{8\sqrt{2}}{3\times 2}}{\frac{\sqrt{6}}{\sqrt{5}}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\frac{3\sqrt{6}}{2}\times \frac{4\sqrt{2}}{3}}{\frac{\sqrt{6}}{\sqrt{5}}}$$

Multiply \frac{3\sqrt{6}}{2} times \frac{4\sqrt{2}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{3\sqrt{6}\times 4\sqrt{2}}{2\times 3}}{\frac{\sqrt{6}}{\sqrt{5}}}$$

Cancel out 2\times 3 in both numerator and denominator.

$$\frac{2\sqrt{2}\sqrt{6}}{\frac{\sqrt{6}}{\sqrt{5}}}$$

Rationalize the denominator of \frac{\sqrt{6}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{2\sqrt{2}\sqrt{6}}{\frac{\sqrt{6}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}$$

The square of \sqrt{5} is 5.

$$\frac{2\sqrt{2}\sqrt{6}}{\frac{\sqrt{6}\sqrt{5}}{5}}$$

To multiply \sqrt{6} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{2\sqrt{2}\sqrt{6}}{\frac{\sqrt{30}}{5}}$$

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}.

$$\frac{2\sqrt{2}\sqrt{2}\sqrt{3}}{\frac{\sqrt{30}}{5}}$$

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

$$\frac{2\times 2\sqrt{3}}{\frac{\sqrt{30}}{5}}$$

Multiply 2 and 2 to get 4.

$$\frac{4\sqrt{3}}{\frac{\sqrt{30}}{5}}$$

Divide 4\sqrt{3} by \frac{\sqrt{30}}{5} by multiplying 4\sqrt{3} by the reciprocal of \frac{\sqrt{30}}{5}.

$$\frac{4\sqrt{3}\times 5}{\sqrt{30}}$$

Rationalize the denominator of \frac{4\sqrt{3}\times 5}{\sqrt{30}} by multiplying numerator and denominator by \sqrt{30}.

$$\frac{4\sqrt{3}\times 5\sqrt{30}}{\left(\sqrt{30}\right)^{2}}$$

The square of \sqrt{30} is 30.

$$\frac{4\sqrt{3}\times 5\sqrt{30}}{30}$$

Multiply 4 and 5 to get 20.

$$\frac{20\sqrt{3}\sqrt{30}}{30}$$

Factor 30=3\times 10. Rewrite the square root of the product \sqrt{3\times 10} as the product of square roots \sqrt{3}\sqrt{10}.

$$\frac{20\sqrt{3}\sqrt{3}\sqrt{10}}{30}$$

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

$$\frac{20\times 3\sqrt{10}}{30}$$

Divide 20\times 3\sqrt{10} by 30 to get \frac{2}{3}\times 3\sqrt{10}.

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

Cancel out 3 and 3.

$$2\sqrt{10}$$