Solve (sqrt{60}+sqrt{15}):sqrt{3}+sqrt{500}-sqrt{800}

$$( \sqrt { 60 } + \sqrt { 15 } ) : \sqrt { 3 } + \sqrt { 500 } - \sqrt { 800 }$$

Evaluate

$$13\sqrt{5}-20\sqrt{2}\approx 0.78461246$$

Solution Steps

Factor 60=2^{2}\times 15. Rewrite the square root of the product \sqrt{2^{2}\times 15} as the product of square roots \sqrt{2^{2}}\sqrt{15}. Take the square root of 2^{2}.

$$\frac{2\sqrt{15}+\sqrt{15}}{\sqrt{3}}+\sqrt{500}-\sqrt{800}$$

Combine 2\sqrt{15} and \sqrt{15} to get 3\sqrt{15}.

$$\frac{3\sqrt{15}}{\sqrt{3}}+\sqrt{500}-\sqrt{800}$$

Rationalize the denominator of \frac{3\sqrt{15}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{3\sqrt{15}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\sqrt{500}-\sqrt{800}$$

The square of \sqrt{3} is 3.

$$\frac{3\sqrt{15}\sqrt{3}}{3}+\sqrt{500}-\sqrt{800}$$

Factor 15=3\times 5. Rewrite the square root of the product \sqrt{3\times 5} as the product of square roots \sqrt{3}\sqrt{5}.

$$\frac{3\sqrt{3}\sqrt{5}\sqrt{3}}{3}+\sqrt{500}-\sqrt{800}$$

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

$$\frac{3\times 3\sqrt{5}}{3}+\sqrt{500}-\sqrt{800}$$

Cancel out 3 and 3.

$$3\sqrt{5}+\sqrt{500}-\sqrt{800}$$

Factor 500=10^{2}\times 5. Rewrite the square root of the product \sqrt{10^{2}\times 5} as the product of square roots \sqrt{10^{2}}\sqrt{5}. Take the square root of 10^{2}.

$$3\sqrt{5}+10\sqrt{5}-\sqrt{800}$$

Combine 3\sqrt{5} and 10\sqrt{5} to get 13\sqrt{5}.

$$13\sqrt{5}-\sqrt{800}$$

Factor 800=20^{2}\times 2. Rewrite the square root of the product \sqrt{20^{2}\times 2} as the product of square roots \sqrt{20^{2}}\sqrt{2}. Take the square root of 20^{2}.

$$13\sqrt{5}-20\sqrt{2}$$