Solve frac{sqrt{20}+sqrt{45}}{sqrt{10}}

$$\frac { \sqrt { 20 } + \sqrt { 45 } } { \sqrt { 10 } }$$

Evaluate

$$\frac{5\sqrt{2}}{2}\approx 3.535533906$$

Solution Steps

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\frac{2\sqrt{5}+\sqrt{45}}{\sqrt{10}}$$

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$\frac{2\sqrt{5}+3\sqrt{5}}{\sqrt{10}}$$

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

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

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

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

The square of \sqrt{10} is 10.

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

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}.

$$\frac{5\sqrt{5}\sqrt{5}\sqrt{2}}{10}$$

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

$$\frac{5\times 5\sqrt{2}}{10}$$

Multiply 5 and 5 to get 25.

$$\frac{25\sqrt{2}}{10}$$

Divide 25\sqrt{2} by 10 to get \frac{5}{2}\sqrt{2}.

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