# How do you find the real and imaginary part 12i^12 + pi(i)?

Mar 22, 2016

$12 {i}^{12} = 12$ is real. $\pi i$ is imaginary.

#### Explanation:

Note that we can simplify ${i}^{12}$:

$\left\{\begin{matrix}i = \textcolor{red}{\sqrt{- 1}} \\ {i}^{2} = {\left(\textcolor{red}{\sqrt{- 1}}\right)}^{2} = \textcolor{b l u e}{- 1} \\ {i}^{4} = {\left({i}^{2}\right)}^{2} = {\left(\textcolor{b l u e}{- 1}\right)}^{2} = \textcolor{g r e e n}{1} \\ {i}^{12} = {\left({i}^{4}\right)}^{3} = {\textcolor{g r e e n}{1}}^{3} = 1\end{matrix}\right.$

Thus, the expression simplifies to be:

$12 {i}^{12} + \pi i = 12 \left(1\right) + \pi i = 12 + \pi i$

This is a complex number in the form

$a + b i$

where $a = 12$ and $b = \pi$. Since $\pi$ is multiplied by $i$, the imaginary unit, $\pi i$ is the imaginary part of the expression. So, $12 {i}^{12} = 12$ is the real part.