Solve 5*(1/2sqrt{5}-1/5sqrt{7})-4*(frac{sqrt{5}}{2}-frac{sqrt{7}}{4})+sqrt{5}

$$5 \times ( \frac{ 1 }{ 2 } \sqrt{ 5 } - \frac{ 1 }{ 5 } \sqrt{ 7 } )-4 \times ( \frac{ \sqrt{ 5 } }{ 2 } - \frac{ \sqrt{ 7 } }{ 4 } )+ \sqrt{ 5 }$$

Evaluate

$$\frac{3\sqrt{5}}{2}\approx 3.354101966$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 4 is 4. Multiply \frac{\sqrt{5}}{2} times \frac{2}{2}.

$$5\left(\frac{1}{2}\sqrt{5}-\frac{1}{5}\sqrt{7}\right)-4\left(\frac{2\sqrt{5}}{4}-\frac{\sqrt{7}}{4}\right)+\sqrt{5}$$

Since \frac{2\sqrt{5}}{4} and \frac{\sqrt{7}}{4} have the same denominator, subtract them by subtracting their numerators.

$$5\left(\frac{1}{2}\sqrt{5}-\frac{1}{5}\sqrt{7}\right)-4\times \frac{2\sqrt{5}-\sqrt{7}}{4}+\sqrt{5}$$

Cancel out 4 and 4.

$$5\left(\frac{1}{2}\sqrt{5}-\frac{1}{5}\sqrt{7}\right)-\left(2\sqrt{5}-\sqrt{7}\right)+\sqrt{5}$$

To find the opposite of 2\sqrt{5}-\sqrt{7}, find the opposite of each term.

$$5\left(\frac{1}{2}\sqrt{5}-\frac{1}{5}\sqrt{7}\right)-2\sqrt{5}-\left(-\sqrt{7}\right)+\sqrt{5}$$

The opposite of -\sqrt{7} is \sqrt{7}.

$$5\left(\frac{1}{2}\sqrt{5}-\frac{1}{5}\sqrt{7}\right)-2\sqrt{5}+\sqrt{7}+\sqrt{5}$$

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

$$5\left(\frac{1}{2}\sqrt{5}-\frac{1}{5}\sqrt{7}\right)-\sqrt{5}+\sqrt{7}$$

Use the distributive property to multiply 5 by \frac{1}{2}\sqrt{5}-\frac{1}{5}\sqrt{7}.

$$5\times \frac{1}{2}\sqrt{5}+5\left(-\frac{1}{5}\right)\sqrt{7}-\sqrt{5}+\sqrt{7}$$

Multiply 5 and \frac{1}{2} to get \frac{5}{2}.

$$\frac{5}{2}\sqrt{5}+5\left(-\frac{1}{5}\right)\sqrt{7}-\sqrt{5}+\sqrt{7}$$

Cancel out 5 and 5.

$$\frac{5}{2}\sqrt{5}-\sqrt{7}-\sqrt{5}+\sqrt{7}$$

Combine \frac{5}{2}\sqrt{5} and -\sqrt{5} to get \frac{3}{2}\sqrt{5}.

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

Combine -\sqrt{7} and \sqrt{7} to get 0.

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