Solve frac{sqrt{75}+sqrt{50}}{sqrt{5}}

$$\frac { \sqrt { 75 } + \sqrt { 50 } } { \sqrt { 5 } }$$

Evaluate

$$\sqrt{10}+\sqrt{15}\approx 7.035261006$$

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}.

$$\frac{5\sqrt{3}+\sqrt{50}}{\sqrt{5}}$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

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

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

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

The square of \sqrt{5} is 5.

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

Use the distributive property to multiply 5\sqrt{3}+5\sqrt{2} by \sqrt{5}.

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

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

$$\frac{5\sqrt{15}+5\sqrt{2}\sqrt{5}}{5}$$

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

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

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

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