Solve 3/sqrt{5}(2sqrt{2}-sqrt{3})(sqrt{2}+2sqrt{3})-sqrt{24}

$$\frac { 3 } { \sqrt { 5 } } ( 2 \sqrt { 2 } - \sqrt { 3 } ) ( \sqrt { 2 } + 2 \sqrt { 3 } ) - \sqrt { 24 }$$

Evaluate

$$\frac{9\sqrt{30}}{5}-\frac{6\sqrt{5}}{5}-2\sqrt{6}\approx 2.276744977$$

Short Solution Steps

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

$$\frac{3\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\left(2\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}+2\sqrt{3}\right)-\sqrt{24}$$

The square of \sqrt{5} is 5.

$$\frac{3\sqrt{5}}{5}\left(2\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}+2\sqrt{3}\right)-\sqrt{24}$$

Express \frac{3\sqrt{5}}{5}\left(2\sqrt{2}-\sqrt{3}\right) as a single fraction.

$$\frac{3\sqrt{5}\left(2\sqrt{2}-\sqrt{3}\right)}{5}\left(\sqrt{2}+2\sqrt{3}\right)-\sqrt{24}$$

Express \frac{3\sqrt{5}\left(2\sqrt{2}-\sqrt{3}\right)}{5}\left(\sqrt{2}+2\sqrt{3}\right) as a single fraction.

$$\frac{3\sqrt{5}\left(2\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}+2\sqrt{3}\right)}{5}-\sqrt{24}$$

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

$$\frac{3\sqrt{5}\left(2\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}+2\sqrt{3}\right)}{5}-2\sqrt{6}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -2\sqrt{6} times \frac{5}{5}.

$$\frac{3\sqrt{5}\left(2\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}+2\sqrt{3}\right)}{5}+\frac{5\left(-2\right)\sqrt{6}}{5}$$

Since \frac{3\sqrt{5}\left(2\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}+2\sqrt{3}\right)}{5} and \frac{5\left(-2\right)\sqrt{6}}{5} have the same denominator, add them by adding their numerators.

$$\frac{3\sqrt{5}\left(2\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}+2\sqrt{3}\right)+5\left(-2\right)\sqrt{6}}{5}$$

Do the multiplications in 3\sqrt{5}\left(2\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}+2\sqrt{3}\right)+5\left(-2\right)\sqrt{6}.

$$\frac{12\sqrt{5}+12\sqrt{30}-3\sqrt{30}-18\sqrt{5}-10\sqrt{6}}{5}$$

Do the calculations in 12\sqrt{5}+12\sqrt{30}-3\sqrt{30}-18\sqrt{5}-10\sqrt{6}.

$$\frac{-6\sqrt{5}-10\sqrt{6}+9\sqrt{30}}{5}$$