Solve frac{sqrt{5}}{sqrt{6}+2}-2/5sqrt{3}

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

Evaluate

$$\frac{\sqrt{30}}{2}-\frac{2\sqrt{3}}{15}-\sqrt{5}\approx 0.271604702$$

Short Solution Steps

Rationalize the denominator of \frac{\sqrt{5}}{\sqrt{6}+2} by multiplying numerator and denominator by \sqrt{6}-2.

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

Consider \left(\sqrt{6}+2\right)\left(\sqrt{6}-2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{6}. Square 2.

$$\frac{\sqrt{5}\left(\sqrt{6}-2\right)}{6-4}-\frac{2}{5\sqrt{3}}$$

Subtract 4 from 6 to get 2.

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 5 and 3 to get 15.

$$\frac{\sqrt{5}\left(\sqrt{6}-2\right)}{2}-\frac{2\sqrt{3}}{15}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 15 is 30. Multiply \frac{\sqrt{5}\left(\sqrt{6}-2\right)}{2} times \frac{15}{15}. Multiply \frac{2\sqrt{3}}{15} times \frac{2}{2}.

$$\frac{15\sqrt{5}\left(\sqrt{6}-2\right)}{30}-\frac{2\times 2\sqrt{3}}{30}$$

Since \frac{15\sqrt{5}\left(\sqrt{6}-2\right)}{30} and \frac{2\times 2\sqrt{3}}{30} have the same denominator, subtract them by subtracting their numerators.

$$\frac{15\sqrt{5}\left(\sqrt{6}-2\right)-2\times 2\sqrt{3}}{30}$$

Do the multiplications in 15\sqrt{5}\left(\sqrt{6}-2\right)-2\times 2\sqrt{3}.

$$\frac{15\sqrt{30}-30\sqrt{5}-4\sqrt{3}}{30}$$

Factor

$$\frac{15 \sqrt{30} - 4 \sqrt{3} - 30 \sqrt{5}}{30} = 0.2716047023501901$$