Solve 40(frac{sqrt{24}}{sqrt{5}}*frac{sqrt{125}}{10sqrt{96}})

$$40 ( \frac { \sqrt { 24 } } { \sqrt { 5 } } \times \frac { \sqrt { 125 } } { 10 \sqrt { 96 } } )$$

Evaluate

$$10$$

Solution Steps

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}.

$$40\times \frac{2\sqrt{6}}{\sqrt{5}}\times \frac{\sqrt{125}}{10\sqrt{96}}$$

Rationalize the denominator of \frac{2\sqrt{6}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$40\times \frac{2\sqrt{6}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\times \frac{\sqrt{125}}{10\sqrt{96}}$$

The square of \sqrt{5} is 5.

$$40\times \frac{2\sqrt{6}\sqrt{5}}{5}\times \frac{\sqrt{125}}{10\sqrt{96}}$$

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

$$40\times \frac{2\sqrt{30}}{5}\times \frac{\sqrt{125}}{10\sqrt{96}}$$

Factor 125=5^{2}\times 5. Rewrite the square root of the product \sqrt{5^{2}\times 5} as the product of square roots \sqrt{5^{2}}\sqrt{5}. Take the square root of 5^{2}.

$$40\times \frac{2\sqrt{30}}{5}\times \frac{5\sqrt{5}}{10\sqrt{96}}$$

Factor 96=4^{2}\times 6. Rewrite the square root of the product \sqrt{4^{2}\times 6} as the product of square roots \sqrt{4^{2}}\sqrt{6}. Take the square root of 4^{2}.

$$40\times \frac{2\sqrt{30}}{5}\times \frac{5\sqrt{5}}{10\times 4\sqrt{6}}$$

Multiply 10 and 4 to get 40.

$$40\times \frac{2\sqrt{30}}{5}\times \frac{5\sqrt{5}}{40\sqrt{6}}$$

Cancel out 5 in both numerator and denominator.

$$40\times \frac{2\sqrt{30}}{5}\times \frac{\sqrt{5}}{8\sqrt{6}}$$

Rationalize the denominator of \frac{\sqrt{5}}{8\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$40\times \frac{2\sqrt{30}}{5}\times \frac{\sqrt{5}\sqrt{6}}{8\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$40\times \frac{2\sqrt{30}}{5}\times \frac{\sqrt{5}\sqrt{6}}{8\times 6}$$

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

$$40\times \frac{2\sqrt{30}}{5}\times \frac{\sqrt{30}}{8\times 6}$$

Multiply 8 and 6 to get 48.

$$40\times \frac{2\sqrt{30}}{5}\times \frac{\sqrt{30}}{48}$$

Cancel out 5, the greatest common factor in 40 and 5.

$$8\times 2\sqrt{30}\times \frac{\sqrt{30}}{48}$$

Express 8\times 2\sqrt{30}\times \frac{\sqrt{30}}{48} as a single fraction.

$$\frac{8\times 2\sqrt{30}\sqrt{30}}{48}$$

Multiply \sqrt{30} and \sqrt{30} to get 30.

$$\frac{8\times 2\times 30}{48}$$

Multiply 8 and 2 to get 16.

$$\frac{16\times 30}{48}$$

Multiply 16 and 30 to get 480.

$$\frac{480}{48}$$

Divide 480 by 48 to get 10.

$$10$$

Factor

$$2\times 5$$