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

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

Evaluate

$$3$$

Short Solution Steps

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2^{2} and 2 is 4. Multiply \frac{\sqrt{5}+1}{2} times \frac{2}{2}.

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

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

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

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

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

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

$$\frac{8}{4}+1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{4}{4}.

$$\frac{8}{4}+\frac{4}{4}$$

Since \frac{8}{4} and \frac{4}{4} have the same denominator, add them by adding their numerators. Add 8 and 4 to get 12.

$$\frac{12}{4}$$

Divide 12 by 4 to get 3.

$$3$$

Factor

$$3$$