Solve {left(frac{-1+sqrt{-3}}{2}right)}^2

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

Evaluate

$$\text{Indeterminate}$$

Solution Steps

To raise \frac{-1+\sqrt{-3}}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-1+\sqrt{-3}\right)^{2}.

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

Calculate \sqrt{-3} to the power of 2 and get -3.

$$\frac{1-2\sqrt{-3}-3}{2^{2}}$$

Subtract 3 from 1 to get -2.

$$\frac{-2-2\sqrt{-3}}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{-2-2\sqrt{-3}}{4}$$

Evaluate (complex solution)

$$\frac{-\sqrt{3}i-1}{2}\approx -0.5-0.866025404i$$

Expand (complex solution)

$$\frac{-\sqrt{3}i-1}{2}$$