Solve (5/2sqrt{5}+frac{2sqrt{5}}{5}+2)*55/2sqrt{75}=

$$( \frac { 5 } { 2 \sqrt { 5 } } + \frac { 2 \sqrt { 5 } } { 5 } + 2 ) \cdot \frac { 55 } { 2 \sqrt { 75 } } =$$

Evaluate

$$\frac{11\sqrt{3}}{3}+\frac{33\sqrt{15}}{20}\approx 12.741275482$$

Short Solution Steps

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

$$\left(\frac{5\sqrt{5}}{2\left(\sqrt{5}\right)^{2}}+\frac{2\sqrt{5}}{5}+2\right)\times \frac{55}{2\sqrt{75}}$$

The square of \sqrt{5} is 5.

$$\left(\frac{5\sqrt{5}}{2\times 5}+\frac{2\sqrt{5}}{5}+2\right)\times \frac{55}{2\sqrt{75}}$$

Cancel out 5 in both numerator and denominator.

$$\left(\frac{\sqrt{5}}{2}+\frac{2\sqrt{5}}{5}+2\right)\times \frac{55}{2\sqrt{75}}$$

Combine \frac{\sqrt{5}}{2} and \frac{2\sqrt{5}}{5} to get \frac{9}{10}\sqrt{5}.

$$\left(\frac{9}{10}\sqrt{5}+2\right)\times \frac{55}{2\sqrt{75}}$$

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$\left(\frac{9}{10}\sqrt{5}+2\right)\times \frac{55}{2\times 5\sqrt{3}}$$

Multiply 2 and 5 to get 10.

$$\left(\frac{9}{10}\sqrt{5}+2\right)\times \frac{55}{10\sqrt{3}}$$

Rationalize the denominator of \frac{55}{10\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\left(\frac{9}{10}\sqrt{5}+2\right)\times \frac{55\sqrt{3}}{10\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\left(\frac{9}{10}\sqrt{5}+2\right)\times \frac{55\sqrt{3}}{10\times 3}$$

Cancel out 5 in both numerator and denominator.

$$\left(\frac{9}{10}\sqrt{5}+2\right)\times \frac{11\sqrt{3}}{2\times 3}$$

Multiply 2 and 3 to get 6.

$$\left(\frac{9}{10}\sqrt{5}+2\right)\times \frac{11\sqrt{3}}{6}$$

Use the distributive property to multiply \frac{9}{10}\sqrt{5}+2 by \frac{11\sqrt{3}}{6}.

$$\frac{9}{10}\sqrt{5}\times \frac{11\sqrt{3}}{6}+2\times \frac{11\sqrt{3}}{6}$$

Multiply \frac{9}{10} times \frac{11\sqrt{3}}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{9\times 11\sqrt{3}}{10\times 6}\sqrt{5}+2\times \frac{11\sqrt{3}}{6}$$

Cancel out 3 in both numerator and denominator.

$$\frac{3\times 11\sqrt{3}}{2\times 10}\sqrt{5}+2\times \frac{11\sqrt{3}}{6}$$

Cancel out 6, the greatest common factor in 2 and 6.

$$\frac{3\times 11\sqrt{3}}{2\times 10}\sqrt{5}+\frac{11\sqrt{3}}{3}$$

Multiply 3 and 11 to get 33.

$$\frac{33\sqrt{3}}{2\times 10}\sqrt{5}+\frac{11\sqrt{3}}{3}$$

Multiply 2 and 10 to get 20.

$$\frac{33\sqrt{3}}{20}\sqrt{5}+\frac{11\sqrt{3}}{3}$$

Express \frac{33\sqrt{3}}{20}\sqrt{5} as a single fraction.

$$\frac{33\sqrt{3}\sqrt{5}}{20}+\frac{11\sqrt{3}}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 20 and 3 is 60. Multiply \frac{33\sqrt{3}\sqrt{5}}{20} times \frac{3}{3}. Multiply \frac{11\sqrt{3}}{3} times \frac{20}{20}.

$$\frac{3\times 33\sqrt{3}\sqrt{5}}{60}+\frac{20\times 11\sqrt{3}}{60}$$

Since \frac{3\times 33\sqrt{3}\sqrt{5}}{60} and \frac{20\times 11\sqrt{3}}{60} have the same denominator, add them by adding their numerators.

$$\frac{3\times 33\sqrt{3}\sqrt{5}+20\times 11\sqrt{3}}{60}$$

Do the multiplications in 3\times 33\sqrt{3}\sqrt{5}+20\times 11\sqrt{3}.

$$\frac{99\sqrt{15}+220\sqrt{3}}{60}$$

Factor

$$\frac{11 \sqrt{3} {(9 \sqrt{5} + 20)}}{60} = 12.74127548232812$$