Solve frac{5sqrt{90}}{3sqrt{5}}=

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

Evaluate

$$5\sqrt{2}\approx 7.071067812$$

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}.

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

Multiply 5 and 3 to get 15.

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

Cancel out 3 in both numerator and denominator.

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

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

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

The square of \sqrt{5} is 5.

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

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}.

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

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

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

Cancel out 5 and 5.

$$5\sqrt{2}$$