Solve frac{sqrt{6}}{2}-9/sqrt{10}divsqrt{22/5}

$$\frac{ \sqrt{ 6 } }{ 2 } - \frac{ 9 }{ \sqrt{ 10 } } \div \sqrt{ 2 \frac{ 2 }{ 5 } }$$

Evaluate

$$-\frac{\sqrt{6}}{4}\approx -0.612372436$$

Solution Steps

Rationalize the denominator of \frac{9}{\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{\sqrt{6}}{2}-\frac{\frac{9\sqrt{10}}{\left(\sqrt{10}\right)^{2}}}{\sqrt{\frac{2\times 5+2}{5}}}$$

The square of \sqrt{10} is 10.

$$\frac{\sqrt{6}}{2}-\frac{\frac{9\sqrt{10}}{10}}{\sqrt{\frac{2\times 5+2}{5}}}$$

Multiply 2 and 5 to get 10.

$$\frac{\sqrt{6}}{2}-\frac{\frac{9\sqrt{10}}{10}}{\sqrt{\frac{10+2}{5}}}$$

Add 10 and 2 to get 12.

$$\frac{\sqrt{6}}{2}-\frac{\frac{9\sqrt{10}}{10}}{\sqrt{\frac{12}{5}}}$$

Rewrite the square root of the division \sqrt{\frac{12}{5}} as the division of square roots \frac{\sqrt{12}}{\sqrt{5}}.

$$\frac{\sqrt{6}}{2}-\frac{\frac{9\sqrt{10}}{10}}{\frac{\sqrt{12}}{\sqrt{5}}}$$

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}.

$$\frac{\sqrt{6}}{2}-\frac{\frac{9\sqrt{10}}{10}}{\frac{2\sqrt{3}}{\sqrt{5}}}$$

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

$$\frac{\sqrt{6}}{2}-\frac{\frac{9\sqrt{10}}{10}}{\frac{2\sqrt{3}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{6}}{2}-\frac{\frac{9\sqrt{10}}{10}}{\frac{2\sqrt{3}\sqrt{5}}{5}}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\sqrt{6}}{2}-\frac{\frac{9\sqrt{10}}{10}}{\frac{2\sqrt{15}}{5}}$$

Divide \frac{9\sqrt{10}}{10} by \frac{2\sqrt{15}}{5} by multiplying \frac{9\sqrt{10}}{10} by the reciprocal of \frac{2\sqrt{15}}{5}.

$$\frac{\sqrt{6}}{2}-\frac{9\sqrt{10}\times 5}{10\times 2\sqrt{15}}$$

Cancel out 5 in both numerator and denominator.

$$\frac{\sqrt{6}}{2}-\frac{9\sqrt{10}}{2\times 2\sqrt{15}}$$

Rationalize the denominator of \frac{9\sqrt{10}}{2\times 2\sqrt{15}} by multiplying numerator and denominator by \sqrt{15}.

$$\frac{\sqrt{6}}{2}-\frac{9\sqrt{10}\sqrt{15}}{2\times 2\left(\sqrt{15}\right)^{2}}$$

The square of \sqrt{15} is 15.

$$\frac{\sqrt{6}}{2}-\frac{9\sqrt{10}\sqrt{15}}{2\times 2\times 15}$$

To multiply \sqrt{10} and \sqrt{15}, multiply the numbers under the square root.

$$\frac{\sqrt{6}}{2}-\frac{9\sqrt{150}}{2\times 2\times 15}$$

Multiply 2 and 2 to get 4.

$$\frac{\sqrt{6}}{2}-\frac{9\sqrt{150}}{4\times 15}$$

Multiply 4 and 15 to get 60.

$$\frac{\sqrt{6}}{2}-\frac{9\sqrt{150}}{60}$$

Factor 150=5^{2}\times 6. Rewrite the square root of the product \sqrt{5^{2}\times 6} as the product of square roots \sqrt{5^{2}}\sqrt{6}. Take the square root of 5^{2}.

$$\frac{\sqrt{6}}{2}-\frac{9\times 5\sqrt{6}}{60}$$

Multiply 9 and 5 to get 45.

$$\frac{\sqrt{6}}{2}-\frac{45\sqrt{6}}{60}$$

Divide 45\sqrt{6} by 60 to get \frac{3}{4}\sqrt{6}.

$$\frac{\sqrt{6}}{2}-\frac{3}{4}\sqrt{6}$$

Combine \frac{\sqrt{6}}{2} and -\frac{3}{4}\sqrt{6} to get -\frac{1}{4}\sqrt{6}.

$$-\frac{1}{4}\sqrt{6}$$