Solve frac{sqrt{125}(sqrt{75}+sqrt{12})}{sqrt{15}}

$$\frac { \sqrt { 125 } ( \sqrt { 75 } + \sqrt { 12 } ) } { \sqrt { 15 } }$$

Evaluate

$$35$$

Solution Steps

Factor 125=5^{2}\times 5. Rewrite the square root of the product \sqrt{5^{2}\times 5} as the product of square roots \sqrt{5^{2}}\sqrt{5}. Take the square root of 5^{2}.

$$\frac{5\sqrt{5}\left(\sqrt{75}+\sqrt{12}\right)}{\sqrt{15}}$$

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$\frac{5\sqrt{5}\left(5\sqrt{3}+\sqrt{12}\right)}{\sqrt{15}}$$

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}.

$$\frac{5\sqrt{5}\left(5\sqrt{3}+2\sqrt{3}\right)}{\sqrt{15}}$$

Combine 5\sqrt{3} and 2\sqrt{3} to get 7\sqrt{3}.

$$\frac{5\sqrt{5}\times 7\sqrt{3}}{\sqrt{15}}$$

Multiply 5 and 7 to get 35.

$$\frac{35\sqrt{5}\sqrt{3}}{\sqrt{15}}$$

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

$$\frac{35\sqrt{15}}{\sqrt{15}}$$

Rationalize the denominator of \frac{35\sqrt{15}}{\sqrt{15}} by multiplying numerator and denominator by \sqrt{15}.

$$\frac{35\sqrt{15}\sqrt{15}}{\left(\sqrt{15}\right)^{2}}$$

The square of \sqrt{15} is 15.

$$\frac{35\sqrt{15}\sqrt{15}}{15}$$

Multiply \sqrt{15} and \sqrt{15} to get 15.

$$\frac{35\times 15}{15}$$

Multiply 35 and 15 to get 525.

$$\frac{525}{15}$$

Divide 525 by 15 to get 35.

$$35$$

Factor

$$5\times 7$$