Solve frac{sqrt{75}+sqrt{30}}{sqrt{3}}

$$\frac { \sqrt { 75 } + \sqrt { 30 } } { \sqrt { 3 } }$$

Evaluate

$$\sqrt{10}+5\approx 8.16227766$$

Solution Steps

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}.

$$\frac{5\sqrt{3}+\sqrt{30}}{\sqrt{3}}$$

Rationalize the denominator of \frac{5\sqrt{3}+\sqrt{30}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\left(5\sqrt{3}+\sqrt{30}\right)\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(5\sqrt{3}+\sqrt{30}\right)\sqrt{3}}{3}$$

Use the distributive property to multiply 5\sqrt{3}+\sqrt{30} by \sqrt{3}.

$$\frac{5\left(\sqrt{3}\right)^{2}+\sqrt{30}\sqrt{3}}{3}$$

The square of \sqrt{3} is 3.

$$\frac{5\times 3+\sqrt{30}\sqrt{3}}{3}$$

Multiply 5 and 3 to get 15.

$$\frac{15+\sqrt{30}\sqrt{3}}{3}$$

Factor 30=3\times 10. Rewrite the square root of the product \sqrt{3\times 10} as the product of square roots \sqrt{3}\sqrt{10}.

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

Multiply \sqrt{3} and \sqrt{3} to get 3.

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

Divide each term of 15+3\sqrt{10} by 3 to get 5+\sqrt{10}.

$$5+\sqrt{10}$$