Answer
$(\displaystyle \frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i )^{2}=i$
So, $\displaystyle \frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i$ is a square root of $i$.
Work Step by Step
$(\displaystyle \frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i )^{2}\qquad$...Combine the fractions
$=(\displaystyle \frac{\sqrt{2}+\sqrt{2}i}{2})^{2}\qquad$...Apply exponent rule: $(\displaystyle \frac{a}{b})^{n}=\frac{a^{n}}{b^{n}}$
$=\displaystyle \frac{(\sqrt{2}+\sqrt{2}i)^{2}}{2^{2}}\qquad$...Apply Perfect Square Formula: $(a+b)^{2}=a^{2}+2ab+b^{2},a=\sqrt{2},b=\sqrt{2}i$
$=\displaystyle \frac{(\sqrt{2})^{2}+2\sqrt{2}\sqrt{2}i+(\sqrt{2}i)^{2}}{4}$
$=\displaystyle \frac{2+2\cdot 2i+2i^{2}}{4}\qquad$...Apply imaginary number rule: $\quad i^{2}=-1$
$=\displaystyle \frac{2+4i+2\cdot(-1)}{4}\qquad$...Simplify.
$=\displaystyle \frac{4i}{4}$
$=i$
Therefore, $(\displaystyle \frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i )^{2}=i$
