Solve (3sqrt{5}-1)+sqrt{5}(1/sqrt{5}-1)

$$( 3 \sqrt { 5 } - 1 ) + \sqrt { 5 } ( \frac { 1 } { \sqrt { 5 } } - 1 )$$

Evaluate

$$2\sqrt{5}\approx 4.472135955$$

Short Solution Steps

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

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

The square of \sqrt{5} is 5.

$$3\sqrt{5}-1+\sqrt{5}\left(\frac{\sqrt{5}}{5}-1\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{5}{5}.

$$3\sqrt{5}-1+\sqrt{5}\left(\frac{\sqrt{5}}{5}-\frac{5}{5}\right)$$

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

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

Express \sqrt{5}\times \frac{\sqrt{5}-5}{5} as a single fraction.

$$3\sqrt{5}-1+\frac{\sqrt{5}\left(\sqrt{5}-5\right)}{5}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3\sqrt{5}-1 times \frac{5}{5}.

$$\frac{5\left(3\sqrt{5}-1\right)}{5}+\frac{\sqrt{5}\left(\sqrt{5}-5\right)}{5}$$

Since \frac{5\left(3\sqrt{5}-1\right)}{5} and \frac{\sqrt{5}\left(\sqrt{5}-5\right)}{5} have the same denominator, add them by adding their numerators.

$$\frac{5\left(3\sqrt{5}-1\right)+\sqrt{5}\left(\sqrt{5}-5\right)}{5}$$

Do the multiplications in 5\left(3\sqrt{5}-1\right)+\sqrt{5}\left(\sqrt{5}-5\right).

$$\frac{15\sqrt{5}-5+5-5\sqrt{5}}{5}$$

Do the calculations in 15\sqrt{5}-5+5-5\sqrt{5}.

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

Divide 10\sqrt{5} by 5 to get 2\sqrt{5}.

$$2\sqrt{5}$$