What is i to the power of 1/2 I am new to complex numbers and wondered, what is $i^{(1/2)}$ and how do I actually create a complex number, so that $i$ has a different power than $1$ in tuple form, like $(a, b) = a + b*i^2$? Is it just $(a, bi)$?
In the complex plane, $i$ is on the unit circle at an angle of $\frac{\pi}{2}$. Thus it has two square roots, which are obtained by taking half the angle and accounting for wraparound by $2\pi$.

Thus, the most obvious square root is the point on the unit circle with angle $\frac{\pi}{4}$. The less obvious square root is obtained by replacing $\frac{\pi}{2}$ with the equivalent angle $2\pi+\frac{\pi}{2}$, and taking half of it results in $\frac{5\pi}{4}$. Note that the second square root is diametrically opposite from the first square root on the unit circle.

To obtain the coordinates (real and imaginary parts) of these two square roots, you can use the fact that a point at angle $\theta$ on the unit circle has coordinates $(\cos \theta,\sin\theta)$ and substituting the values $\theta=\frac{\pi}{4}$ and $\theta=\frac{5\pi}{4}$ into this.
In reverse order:

$$a+bi^2 = a + b(-1) =  (a-b) + 0i = (a-b, 0)$$

Since the tuple form comes from the standard form of the complex number, there never is a higher power on the $i$ than $1.$

As for powers of $i$... We can find two possibilities for $i^{1/2}$ just by solving 
$$(a+bi)^2 = i \\ (a^2-b^2) + (2ab)i = i \\ \begin{cases}a^2-b^2 = 0 \\ 2ab = 1\end{cases} \\ a = b = \pm \frac{\sqrt{2}}{2} \\ a+bi = \pm\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i\right)$$

A word of warning: While the powers of positive real numbers can be defined as a single number, you will almost certainly run into problems if you want a consistent definition of powers of complex numbers.  In the most general case, the power $(a+bi)^r$ is evaluated as $e^{r\ln(a+bi)},$ which means in order to find a power of a complex number, we first need to define the natural logarithm, which requires a lot of care.