Solve 9/9/4{5^2sqrt{frac{4/frac{8{5}}}}}

$$\frac{ \frac{ 9 }{ 9 } }{ \frac{ 4 }{ { 5 }^{ 2 } \sqrt{ \frac{ 4 }{ \frac{ 8 }{ 5 } } } } }$$

Evaluate

$$\frac{25\sqrt{10}}{8}\approx 9.882117688$$

Solution Steps

Divide 9 by 9 to get 1.

$$\frac{1}{\frac{4}{5^{2}\sqrt{\frac{4}{\frac{8}{5}}}}}$$

Divide 1 by \frac{4}{5^{2}\sqrt{\frac{4}{\frac{8}{5}}}} by multiplying 1 by the reciprocal of \frac{4}{5^{2}\sqrt{\frac{4}{\frac{8}{5}}}}.

$$\frac{5^{2}\sqrt{\frac{4}{\frac{8}{5}}}}{4}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{25\sqrt{\frac{4}{\frac{8}{5}}}}{4}$$

Divide 4 by \frac{8}{5} by multiplying 4 by the reciprocal of \frac{8}{5}.

$$\frac{25\sqrt{4\times \frac{5}{8}}}{4}$$

Express 4\times \frac{5}{8} as a single fraction.

$$\frac{25\sqrt{\frac{4\times 5}{8}}}{4}$$

Multiply 4 and 5 to get 20.

$$\frac{25\sqrt{\frac{20}{8}}}{4}$$

Reduce the fraction \frac{20}{8} to lowest terms by extracting and canceling out 4.

$$\frac{25\sqrt{\frac{5}{2}}}{4}$$

Rewrite the square root of the division \sqrt{\frac{5}{2}} as the division of square roots \frac{\sqrt{5}}{\sqrt{2}}.

$$\frac{25\times \frac{\sqrt{5}}{\sqrt{2}}}{4}$$

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

$$\frac{25\times \frac{\sqrt{5}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}}{4}$$

The square of \sqrt{2} is 2.

$$\frac{25\times \frac{\sqrt{5}\sqrt{2}}{2}}{4}$$

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

$$\frac{25\times \frac{\sqrt{10}}{2}}{4}$$

Express 25\times \frac{\sqrt{10}}{2} as a single fraction.

$$\frac{\frac{25\sqrt{10}}{2}}{4}$$

Express \frac{\frac{25\sqrt{10}}{2}}{4} as a single fraction.

$$\frac{25\sqrt{10}}{2\times 4}$$

Multiply 2 and 4 to get 8.

$$\frac{25\sqrt{10}}{8}$$