Solve 3/5sqrt{1frac{1/42^2}}{3/2sqrt{125*21/2}}

$$\frac{ \frac{ 3 }{ 5 } \sqrt{ 1 \frac{ 1 }{ 4 } { 2 }^{ 2 } } }{ \frac{ 3 }{ 2 } \sqrt{ 125 \cdot 2 \frac{ 1 }{ 2 } } }$$

Evaluate

$$\frac{2\sqrt{10}}{125}\approx 0.050596443$$

Solution Steps

Multiply 1 and 4 to get 4.

$$\frac{\frac{3}{5}\sqrt{\frac{4+1}{4}\times 2^{2}}}{\frac{3}{2}\sqrt{125\times \frac{2\times 2+1}{2}}}$$

Add 4 and 1 to get 5.

$$\frac{\frac{3}{5}\sqrt{\frac{5}{4}\times 2^{2}}}{\frac{3}{2}\sqrt{125\times \frac{2\times 2+1}{2}}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\frac{3}{5}\sqrt{\frac{5}{4}\times 4}}{\frac{3}{2}\sqrt{125\times \frac{2\times 2+1}{2}}}$$

Cancel out 4 and 4.

$$\frac{\frac{3}{5}\sqrt{5}}{\frac{3}{2}\sqrt{125\times \frac{2\times 2+1}{2}}}$$

Multiply 2 and 2 to get 4.

$$\frac{\frac{3}{5}\sqrt{5}}{\frac{3}{2}\sqrt{125\times \frac{4+1}{2}}}$$

Add 4 and 1 to get 5.

$$\frac{\frac{3}{5}\sqrt{5}}{\frac{3}{2}\sqrt{125\times \frac{5}{2}}}$$

Express 125\times \frac{5}{2} as a single fraction.

$$\frac{\frac{3}{5}\sqrt{5}}{\frac{3}{2}\sqrt{\frac{125\times 5}{2}}}$$

Multiply 125 and 5 to get 625.

$$\frac{\frac{3}{5}\sqrt{5}}{\frac{3}{2}\sqrt{\frac{625}{2}}}$$

Rewrite the square root of the division \sqrt{\frac{625}{2}} as the division of square roots \frac{\sqrt{625}}{\sqrt{2}}.

$$\frac{\frac{3}{5}\sqrt{5}}{\frac{3}{2}\times \frac{\sqrt{625}}{\sqrt{2}}}$$

Calculate the square root of 625 and get 25.

$$\frac{\frac{3}{5}\sqrt{5}}{\frac{3}{2}\times \frac{25}{\sqrt{2}}}$$

Rationalize the denominator of \frac{25}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\frac{3}{5}\sqrt{5}}{\frac{3}{2}\times \frac{25\sqrt{2}}{\left(\sqrt{2}\right)^{2}}}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{3}{5}\sqrt{5}}{\frac{3}{2}\times \frac{25\sqrt{2}}{2}}$$

Multiply \frac{3}{2} times \frac{25\sqrt{2}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{3}{5}\sqrt{5}}{\frac{3\times 25\sqrt{2}}{2\times 2}}$$

Divide \frac{3}{5}\sqrt{5} by \frac{3\times 25\sqrt{2}}{2\times 2} by multiplying \frac{3}{5}\sqrt{5} by the reciprocal of \frac{3\times 25\sqrt{2}}{2\times 2}.

$$\frac{\frac{3}{5}\sqrt{5}\times 2\times 2}{3\times 25\sqrt{2}}$$

Rationalize the denominator of \frac{\frac{3}{5}\sqrt{5}\times 2\times 2}{3\times 25\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\frac{3}{5}\sqrt{5}\times 2\times 2\sqrt{2}}{3\times 25\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{3}{5}\sqrt{5}\times 2\times 2\sqrt{2}}{3\times 25\times 2}$$

Express \frac{3}{5}\times 2 as a single fraction.

$$\frac{\frac{3\times 2}{5}\sqrt{5}\times 2\sqrt{2}}{3\times 25\times 2}$$

Multiply 3 and 2 to get 6.

$$\frac{\frac{6}{5}\sqrt{5}\times 2\sqrt{2}}{3\times 25\times 2}$$

Express \frac{6}{5}\times 2 as a single fraction.

$$\frac{\frac{6\times 2}{5}\sqrt{5}\sqrt{2}}{3\times 25\times 2}$$

Multiply 6 and 2 to get 12.

$$\frac{\frac{12}{5}\sqrt{5}\sqrt{2}}{3\times 25\times 2}$$

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

$$\frac{\frac{12}{5}\sqrt{10}}{3\times 25\times 2}$$

Multiply 3 and 25 to get 75.

$$\frac{\frac{12}{5}\sqrt{10}}{75\times 2}$$

Multiply 75 and 2 to get 150.

$$\frac{\frac{12}{5}\sqrt{10}}{150}$$

Divide \frac{12}{5}\sqrt{10} by 150 to get \frac{2}{125}\sqrt{10}.

$$\frac{2}{125}\sqrt{10}$$