Solve 2/sqrt{5}frac{1sqrt{3}}{2}-1/sqrt{5}1/2

$$\frac{ 2 }{ \sqrt{ 5 } } \frac{ 1 \sqrt{ 3 } }{ 2 } - \frac{ 1 }{ \sqrt{ 5 } } \frac{ 1 }{ 2 }$$

Evaluate

$$\frac{\sqrt{15}}{5}-\frac{\sqrt{5}}{10}\approx 0.550989871$$

Short Solution Steps

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

$$\frac{2\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\times \frac{1\sqrt{3}}{2}-\frac{1}{\sqrt{5}}\times \frac{1}{2}$$

The square of \sqrt{5} is 5.

$$\frac{2\sqrt{5}}{5}\times \frac{1\sqrt{3}}{2}-\frac{1}{\sqrt{5}}\times \frac{1}{2}$$

Multiply \frac{2\sqrt{5}}{5} times \frac{1\sqrt{3}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\sqrt{5}\times 1\sqrt{3}}{5\times 2}-\frac{1}{\sqrt{5}}\times \frac{1}{2}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\sqrt{3}\sqrt{5}}{5}-\frac{1}{\sqrt{5}}\times \frac{1}{2}$$

Rationalize the denominator of \frac{1}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\sqrt{3}\sqrt{5}}{5}-\frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\times \frac{1}{2}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{3}\sqrt{5}}{5}-\frac{\sqrt{5}}{5}\times \frac{1}{2}$$

Multiply \frac{\sqrt{5}}{5} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{3}\sqrt{5}}{5}-\frac{\sqrt{5}}{5\times 2}$$

Multiply 5 and 2 to get 10.

$$\frac{\sqrt{3}\sqrt{5}}{5}-\frac{\sqrt{5}}{10}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 10 is 10. Multiply \frac{\sqrt{3}\sqrt{5}}{5} times \frac{2}{2}.

$$\frac{2\sqrt{3}\sqrt{5}}{10}-\frac{\sqrt{5}}{10}$$

Since \frac{2\sqrt{3}\sqrt{5}}{10} and \frac{\sqrt{5}}{10} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\sqrt{3}\sqrt{5}-\sqrt{5}}{10}$$

Do the multiplications in 2\sqrt{3}\sqrt{5}-\sqrt{5}.

$$\frac{2\sqrt{15}-\sqrt{5}}{10}$$

Factor

$$\frac{\sqrt{5} {(2 \sqrt{3} - 1)}}{10} = 0.5509898714915045$$