Solve frac{frac{sqrt{8}}{sqrt{12}}}{sqrt{45/12}}sqrt{5/8}

$$\frac{ \frac{ \sqrt{ 8 } }{ \sqrt{ 12 } } }{ \sqrt{ \frac{ 45 }{ 12 } } } \sqrt{ \frac{ 5 }{ 8 } }$$

Evaluate

$$\frac{1}{3}\approx 0.333333333$$

Solution Steps

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{\frac{2\sqrt{2}}{\sqrt{12}}}{\sqrt{\frac{45}{12}}}\sqrt{\frac{5}{8}}$$

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}.

$$\frac{\frac{2\sqrt{2}}{2\sqrt{3}}}{\sqrt{\frac{45}{12}}}\sqrt{\frac{5}{8}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\frac{\sqrt{2}}{\sqrt{3}}}{\sqrt{\frac{45}{12}}}\sqrt{\frac{5}{8}}$$

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

$$\frac{\frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{\sqrt{\frac{45}{12}}}\sqrt{\frac{5}{8}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{\sqrt{2}\sqrt{3}}{3}}{\sqrt{\frac{45}{12}}}\sqrt{\frac{5}{8}}$$

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

$$\frac{\frac{\sqrt{6}}{3}}{\sqrt{\frac{45}{12}}}\sqrt{\frac{5}{8}}$$

Reduce the fraction \frac{45}{12} to lowest terms by extracting and canceling out 3.

$$\frac{\frac{\sqrt{6}}{3}}{\sqrt{\frac{15}{4}}}\sqrt{\frac{5}{8}}$$

Rewrite the square root of the division \sqrt{\frac{15}{4}} as the division of square roots \frac{\sqrt{15}}{\sqrt{4}}.

$$\frac{\frac{\sqrt{6}}{3}}{\frac{\sqrt{15}}{\sqrt{4}}}\sqrt{\frac{5}{8}}$$

Calculate the square root of 4 and get 2.

$$\frac{\frac{\sqrt{6}}{3}}{\frac{\sqrt{15}}{2}}\sqrt{\frac{5}{8}}$$

Divide \frac{\sqrt{6}}{3} by \frac{\sqrt{15}}{2} by multiplying \frac{\sqrt{6}}{3} by the reciprocal of \frac{\sqrt{15}}{2}.

$$\frac{\sqrt{6}\times 2}{3\sqrt{15}}\sqrt{\frac{5}{8}}$$

Rationalize the denominator of \frac{\sqrt{6}\times 2}{3\sqrt{15}} by multiplying numerator and denominator by \sqrt{15}.

$$\frac{\sqrt{6}\times 2\sqrt{15}}{3\left(\sqrt{15}\right)^{2}}\sqrt{\frac{5}{8}}$$

The square of \sqrt{15} is 15.

$$\frac{\sqrt{6}\times 2\sqrt{15}}{3\times 15}\sqrt{\frac{5}{8}}$$

To multiply \sqrt{6} and \sqrt{15}, multiply the numbers under the square root.

$$\frac{\sqrt{90}\times 2}{3\times 15}\sqrt{\frac{5}{8}}$$

Multiply 3 and 15 to get 45.

$$\frac{\sqrt{90}\times 2}{45}\sqrt{\frac{5}{8}}$$

Factor 90=3^{2}\times 10. Rewrite the square root of the product \sqrt{3^{2}\times 10} as the product of square roots \sqrt{3^{2}}\sqrt{10}. Take the square root of 3^{2}.

$$\frac{3\sqrt{10}\times 2}{45}\sqrt{\frac{5}{8}}$$

Multiply 3 and 2 to get 6.

$$\frac{6\sqrt{10}}{45}\sqrt{\frac{5}{8}}$$

Divide 6\sqrt{10} by 45 to get \frac{2}{15}\sqrt{10}.

$$\frac{2}{15}\sqrt{10}\sqrt{\frac{5}{8}}$$

Rewrite the square root of the division \sqrt{\frac{5}{8}} as the division of square roots \frac{\sqrt{5}}{\sqrt{8}}.

$$\frac{2}{15}\sqrt{10}\times \frac{\sqrt{5}}{\sqrt{8}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{2}{15}\sqrt{10}\times \frac{\sqrt{5}}{2\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{5}}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{2}{15}\sqrt{10}\times \frac{\sqrt{5}\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{2}{15}\sqrt{10}\times \frac{\sqrt{5}\sqrt{2}}{2\times 2}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{2}{15}\sqrt{10}\times \frac{\sqrt{10}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{2}{15}\sqrt{10}\times \frac{\sqrt{10}}{4}$$

Multiply \frac{2}{15} times \frac{\sqrt{10}}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\sqrt{10}}{15\times 4}\sqrt{10}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\sqrt{10}}{2\times 15}\sqrt{10}$$

Multiply 2 and 15 to get 30.

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

Express \frac{\sqrt{10}}{30}\sqrt{10} as a single fraction.

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

Multiply \sqrt{10} and \sqrt{10} to get 10.

$$\frac{10}{30}$$

Reduce the fraction \frac{10}{30} to lowest terms by extracting and canceling out 10.

$$\frac{1}{3}$$

Factor

$$\frac{1}{3} = 0.3333333333333333$$