Solve frac{sqrt{5}}{5}*({left(frac{1+sqrt{5}}{2}right)}^2-{left(frac{1-sqrt{5}}{2}right)}^2)

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

Evaluate

$$1$$

Short Solution Steps

To raise \frac{1+\sqrt{5}}{2} to a power, raise both numerator and denominator to the power and then divide.

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

To raise \frac{1-\sqrt{5}}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-\sqrt{5}\right)^{2}.

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

The square of \sqrt{5} is 5.

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

Add 1 and 5 to get 6.

$$\frac{\sqrt{5}}{5}\left(\frac{\left(1+\sqrt{5}\right)^{2}}{2^{2}}-\frac{6-2\sqrt{5}}{2^{2}}\right)$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\sqrt{5}}{5}\left(\frac{\left(1+\sqrt{5}\right)^{2}}{2^{2}}-\frac{6-2\sqrt{5}}{4}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Expand 2^{2}.

$$\frac{\sqrt{5}}{5}\left(\frac{\left(1+\sqrt{5}\right)^{2}}{4}-\frac{6-2\sqrt{5}}{4}\right)$$

Since \frac{\left(1+\sqrt{5}\right)^{2}}{4} and \frac{6-2\sqrt{5}}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{5}}{5}\times \frac{\left(1+\sqrt{5}\right)^{2}-\left(6-2\sqrt{5}\right)}{4}$$

Do the multiplications in \left(1+\sqrt{5}\right)^{2}-\left(6-2\sqrt{5}\right).

$$\frac{\sqrt{5}}{5}\times \frac{1+2\sqrt{5}+\left(\sqrt{5}\right)^{2}-6+2\sqrt{5}}{4}$$

Do the calculations in 1+2\sqrt{5}+\left(\sqrt{5}\right)^{2}-6+2\sqrt{5}.

$$\frac{\sqrt{5}}{5}\times \frac{4\sqrt{5}}{4}$$

Cancel out 4 and 4.

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

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

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

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{5}{5}$$

Divide 5 by 5 to get 1.

$$1$$

Factor

$$1$$