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

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

Evaluate

$$\frac{\sqrt{5}+3}{2}\approx 2.618033989$$

Solution Steps

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

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

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

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

The square of \sqrt{5} is 5.

$$\frac{1+2\sqrt{5}+5}{2^{2}}$$

Add 1 and 5 to get 6.

$$\frac{6+2\sqrt{5}}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{6+2\sqrt{5}}{4}$$

Expand

$$\frac{\sqrt{5} + 3}{2} = 2.618033988749895$$

Solution Steps

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

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

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

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

The square of \sqrt{5} is 5.

$$\frac{1+2\sqrt{5}+5}{2^{2}}$$

Add 1 and 5 to get 6.

$$\frac{6+2\sqrt{5}}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{6+2\sqrt{5}}{4}$$