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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 $(i^{2}) i$ , which equals $- 1 (i)$ or simply $- i$ .

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

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

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

Recall the definition of $i$ .

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

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

Example 2:

Simplify.

$(2 i) (- 6 i) (- 7 i)$

Rewrite the expression grouping the real and imaginary parts.

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

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 $i^{2} = - 1$ :

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

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.

$= 8 i$

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 $i$ .

Powers of i

Powers of $i$