Solve frac{sqrt{90}}{sqrt{33/5}}

$$\frac{ \sqrt{ 90 } }{ \sqrt{ 3 \frac{ 3 }{ 5 } } }$$

Evaluate

$$5$$

Solution Steps

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}.

$$\frac{3\sqrt{10}}{\sqrt{\frac{3\times 5+3}{5}}}$$

Multiply 3 and 5 to get 15.

$$\frac{3\sqrt{10}}{\sqrt{\frac{15+3}{5}}}$$

Add 15 and 3 to get 18.

$$\frac{3\sqrt{10}}{\sqrt{\frac{18}{5}}}$$

Rewrite the square root of the division \sqrt{\frac{18}{5}} as the division of square roots \frac{\sqrt{18}}{\sqrt{5}}.

$$\frac{3\sqrt{10}}{\frac{\sqrt{18}}{\sqrt{5}}}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$\frac{3\sqrt{10}}{\frac{3\sqrt{2}}{\sqrt{5}}}$$

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

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

The square of \sqrt{5} is 5.

$$\frac{3\sqrt{10}}{\frac{3\sqrt{2}\sqrt{5}}{5}}$$

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

$$\frac{3\sqrt{10}}{\frac{3\sqrt{10}}{5}}$$

Divide 3\sqrt{10} by \frac{3\sqrt{10}}{5} by multiplying 3\sqrt{10} by the reciprocal of \frac{3\sqrt{10}}{5}.

$$\frac{3\sqrt{10}\times 5}{3\sqrt{10}}$$

Cancel out 3\sqrt{10} in both numerator and denominator.

$$5$$

Factor

$$5$$