Imaginary Fractions

Note that $i^7 = -i$ .

The complex conjugate of $-i$ is $i$ .

Then $\frac{2}{i^7} = \frac{2}{-i}$

We multiply the numerator and denominator of the fraction by the conjugate of $-i$ .

So we have:

$\frac{2}{-i} \cdot \frac{i}{i} = \frac{2i}{-i^2} = \frac{2i}{-1(-1)} = 2i$

First, we note that $\dfrac{2}{i^7} = 2i^{-7}$ . Next, we look at the first few whole powers of $i$ . $i^0 = 1$ , $i^1 = i$ , $i^2 =-1$ , $i^3 = -i$ , $i^4 = 1$ , $i^5 = i$ , and so on. We see that the powers of $i$ cycle every fourth power. Next, we consider the value of $i^{-1}$ . This is the same as $\dfrac{i^0}{i} = \dfrac{i^4}{i} = i^3 = -i$ (because $i^0 = i^4$ ). If we do this a few more times, we find that this cycle holds for negative powers as well al positive ones. Therefore, we can just go back through the cycle to find that $i^{-7} = i$ , and multiply by $2$ to get $\fbox{2i}$ .