Solve -1/5sqrt{500}-sqrt{125}+3/2sqrt{8/3}+2sqrt{265}

$$- \frac { 1 } { 5 } \sqrt { 500 } - \sqrt { 125 } + 3 / 2 \sqrt { \frac { 8 } { 3 } } + 2 \sqrt { 265 }$$

Evaluate

$$\sqrt{6}+2\sqrt{265}-7\sqrt{5}\approx 19.354655092$$

Solution Steps

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}.

$$-\frac{1}{5}\times 10\sqrt{5}-\sqrt{125}+\frac{3}{2}\sqrt{\frac{8}{3}}+2\sqrt{265}$$

Express -\frac{1}{5}\times 10 as a single fraction.

$$\frac{-10}{5}\sqrt{5}-\sqrt{125}+\frac{3}{2}\sqrt{\frac{8}{3}}+2\sqrt{265}$$

Divide -10 by 5 to get -2.

$$-2\sqrt{5}-\sqrt{125}+\frac{3}{2}\sqrt{\frac{8}{3}}+2\sqrt{265}$$

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}.

$$-2\sqrt{5}-5\sqrt{5}+\frac{3}{2}\sqrt{\frac{8}{3}}+2\sqrt{265}$$

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

$$-7\sqrt{5}+\frac{3}{2}\sqrt{\frac{8}{3}}+2\sqrt{265}$$

Rewrite the square root of the division \sqrt{\frac{8}{3}} as the division of square roots \frac{\sqrt{8}}{\sqrt{3}}.

$$-7\sqrt{5}+\frac{3}{2}\times \frac{\sqrt{8}}{\sqrt{3}}+2\sqrt{265}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$-7\sqrt{5}+\frac{3}{2}\times \frac{2\sqrt{2}}{\sqrt{3}}+2\sqrt{265}$$

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

$$-7\sqrt{5}+\frac{3}{2}\times \frac{2\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+2\sqrt{265}$$

The square of \sqrt{3} is 3.

$$-7\sqrt{5}+\frac{3}{2}\times \frac{2\sqrt{2}\sqrt{3}}{3}+2\sqrt{265}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$-7\sqrt{5}+\frac{3}{2}\times \frac{2\sqrt{6}}{3}+2\sqrt{265}$$

Multiply \frac{3}{2} times \frac{2\sqrt{6}}{3} by multiplying numerator times numerator and denominator times denominator.

$$-7\sqrt{5}+\frac{3\times 2\sqrt{6}}{2\times 3}+2\sqrt{265}$$

Cancel out 2\times 3 in both numerator and denominator.

$$-7\sqrt{5}+\sqrt{6}+2\sqrt{265}$$

Factor

$$\sqrt{6} + 2 \sqrt{265} - 7 \sqrt{5} = 19.354655092$$