# Powers of $i$

The imaginary unit $i$ is defined as the square root of $-1$. So, ${i}^{2}=-1$.

${i}^{3}$ can be written as $\left({i}^{2}\right)i$, which equals $-1\left(i\right)$ or simply $-i$.

${i}^{4}$ can be written as $\left({i}^{2}\right)\left({i}^{2}\right)$, which equals $\left(-1\right)\left(-1\right)$ or $1$.

${i}^{5}$ can be written as $\left({i}^{4}\right)i$, which equals $\left(1\right)i$ or $i$.

Therefore, the cycle repeats every four powers, as shown in the table.

 Powers of $10$ ${i}^{1}=i$ ${i}^{0}=1$ ${i}^{2}=-1$ ${i}^{-1}=-i$ ${i}^{3}=-i$ ${i}^{-2}=-1$ ${i}^{4}=1$ ${i}^{-3}=i$ ${i}^{5}=i$ ${i}^{-4}=1$ ${i}^{6}=-1$ ${i}^{-5}=-i$ ${i}^{7}=-i$ ${i}^{-6}=-1$ ${i}^{8}=1$ ${i}^{-7}=i$ ${i}^{9}=i$ ${i}^{-8}=1$ etc. etc.

Example 1:

Simplify.

$-5{i}^{4}$

Simplify the imaginary part using the property of multiplying powers.

$-5{i}^{4}=\left(-5\right)\cdot {i}^{2}\cdot {i}^{2}$

Recall the definition of $i$.

Since ${i}^{2}=-1$:

$\begin{array}{l}=\left(-5\right)\cdot \left(-1\right)\cdot \left(-1\right)\\ =-5\end{array}$

Example 2:

Simplify.

$\left(2i\right)\left(-6i\right)\left(-7i\right)$

Rewrite the expression grouping the real and imaginary parts.

$\begin{array}{l}\left(2i\right)\left(-6i\right)\left(-7i\right)=\left(2\right)\cdot \left(-6\right)\cdot \left(-7\right)\cdot i\cdot i\cdot i\\ =84\cdot i\cdot i\cdot i\end{array}$

Simplify the imaginary part using the property of multiplying powers.

$\begin{array}{l}=84{i}^{3}\\ =84{i}^{2}{i}^{1}\end{array}$

Recall the definition of $i$.

Since ${i}^{2}=-1$:

$\begin{array}{l}=84\left(-1\right)i\\ =-84i\end{array}$

Example 3:

Simplify the principal square roots.

$\sqrt{-64}$

Taking the square root and substituting $\sqrt{-1}=i$:

$\begin{array}{l}=\sqrt{-1}\cdot \sqrt{64}\\ =i\sqrt{64}\end{array}$

Simplify.

$=8i$

Example 4:

Simplify the principal square roots.

$\sqrt{-121{x}^{4}}$

Factor the radicand into squares,

$\begin{array}{l}\sqrt{-121{x}^{4}}=\sqrt{-1\cdot 121\cdot {x}^{4}}\\ =11{x}^{2}\sqrt{-1}\end{array}$

Rewrite the expression using $i$.

$=11{x}^{2}i$

Note: When you are simplifying radicals as part of an equation, please remember that unless a principal square root is requested, there are always both positive and negative roots - including when you are working with $i$.