Solve -sqrt{90}divsqrt{33/5}

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

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}.

$$\frac{\left(-3\sqrt{10}\right)\times 5}{3\sqrt{10}}$$

Rationalize the denominator of \frac{\left(-3\sqrt{10}\right)\times 5}{3\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

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

The square of \sqrt{10} is 10.

$$\frac{\left(-3\sqrt{10}\right)\times 5\sqrt{10}}{3\times 10}$$

Multiply 3 and 10 to get 30.

$$\frac{\left(-3\sqrt{10}\right)\times 5\sqrt{10}}{30}$$

Divide \left(-3\sqrt{10}\right)\times 5\sqrt{10} by 30 to get \left(-3\sqrt{10}\right)\times \frac{1}{6}\sqrt{10}.

$$\left(-3\sqrt{10}\right)\times \frac{1}{6}\sqrt{10}$$

Multiply -1 and 3 to get -3.

$$-3\sqrt{10}\times \frac{1}{6}\sqrt{10}$$

Multiply -3 and \frac{1}{6} to get \frac{-3}{6}.

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

Reduce the fraction \frac{-3}{6} to lowest terms by extracting and canceling out 3.

$$-\frac{1}{2}\sqrt{10}\sqrt{10}$$

Multiply \sqrt{10} and \sqrt{10} to get 10.

$$-\frac{1}{2}\times 10$$

Express -\frac{1}{2}\times 10 as a single fraction.

$$\frac{-10}{2}$$

Divide -10 by 2 to get -5.

$$-5$$

Factor

$$-5$$