Solve frac{sqrt{5}+1/sqrt{5}}{sqrt{10}-4/sqrt{90}}

$$\frac { \sqrt { 5 } + \frac { 1 } { \sqrt { 5 } } } { \sqrt { 10 } - \frac { 4 } { \sqrt { 90 } } }$$

Evaluate

$$\frac{9\sqrt{2}}{13}\approx 0.979070928$$

Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\sqrt{5}+\frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}{\sqrt{10}-\frac{4}{\sqrt{90}}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{5}+\frac{\sqrt{5}}{5}}{\sqrt{10}-\frac{4}{\sqrt{90}}}$$

Combine \sqrt{5} and \frac{\sqrt{5}}{5} to get \frac{6}{5}\sqrt{5}.

$$\frac{\frac{6}{5}\sqrt{5}}{\sqrt{10}-\frac{4}{\sqrt{90}}}$$

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}.

$$\frac{\frac{6}{5}\sqrt{5}}{\sqrt{10}-\frac{4}{3\sqrt{10}}}$$

Rationalize the denominator of \frac{4}{3\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{\frac{6}{5}\sqrt{5}}{\sqrt{10}-\frac{4\sqrt{10}}{3\left(\sqrt{10}\right)^{2}}}$$

The square of \sqrt{10} is 10.

$$\frac{\frac{6}{5}\sqrt{5}}{\sqrt{10}-\frac{4\sqrt{10}}{3\times 10}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\frac{6}{5}\sqrt{5}}{\sqrt{10}-\frac{2\sqrt{10}}{3\times 5}}$$

Multiply 3 and 5 to get 15.

$$\frac{\frac{6}{5}\sqrt{5}}{\sqrt{10}-\frac{2\sqrt{10}}{15}}$$

Combine \sqrt{10} and -\frac{2\sqrt{10}}{15} to get \frac{13}{15}\sqrt{10}.

$$\frac{\frac{6}{5}\sqrt{5}}{\frac{13}{15}\sqrt{10}}$$

Rationalize the denominator of \frac{\frac{6}{5}\sqrt{5}}{\frac{13}{15}\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{\frac{6}{5}\sqrt{5}\sqrt{10}}{\frac{13}{15}\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{\frac{6}{5}\sqrt{5}\sqrt{10}}{\frac{13}{15}\times 10}$$

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

$$\frac{\frac{6}{5}\sqrt{5}\sqrt{5}\sqrt{2}}{\frac{13}{15}\times 10}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{\frac{6}{5}\times 5\sqrt{2}}{\frac{13}{15}\times 10}$$

Express \frac{13}{15}\times 10 as a single fraction.

$$\frac{\frac{6}{5}\times 5\sqrt{2}}{\frac{13\times 10}{15}}$$

Multiply 13 and 10 to get 130.

$$\frac{\frac{6}{5}\times 5\sqrt{2}}{\frac{130}{15}}$$

Reduce the fraction \frac{130}{15} to lowest terms by extracting and canceling out 5.

$$\frac{\frac{6}{5}\times 5\sqrt{2}}{\frac{26}{3}}$$

Cancel out 5 and 5.

$$\frac{6\sqrt{2}}{\frac{26}{3}}$$

Divide 6\sqrt{2} by \frac{26}{3} by multiplying 6\sqrt{2} by the reciprocal of \frac{26}{3}.

$$\frac{6\sqrt{2}\times 3}{26}$$

Divide 6\sqrt{2}\times 3 by 26 to get \frac{3}{13}\sqrt{2}\times 3.

$$\frac{3}{13}\sqrt{2}\times 3$$

Express \frac{3}{13}\times 3 as a single fraction.

$$\frac{3\times 3}{13}\sqrt{2}$$

Multiply 3 and 3 to get 9.

$$\frac{9}{13}\sqrt{2}$$