In Example $1,$ $(2 i)^{n}$ was computed for $n = 0,$ $1,$ $2,$ and $3 .$

$n$	$(2 i)^{n}$
$0$	$1$
$1$	$2 i$
$2$	$- 4$
$3$	$- 8 i$

Additionally, these points where plotted on the complex plane.

Our mission is to predict the location of $(2 i)^{5} .$ To do this, we will make a table of values where we point out some important facts from the graph above.

$n$	$(2 i)^{n}$	Location	Distance to the Origin
$0$	$1$	Real-axis	$1$
$1$	$2 i$	Imaginary-axis	$2$
$2$	$- 4$	Real-axis	$4$
$3$	$- 8 i$	Imaginary-axis	$8$

The table above leads us to say that $(2 i)^{4}$ will be on the real-axis, so $(2 i)^{5}$ will lie on the imaginary-axis. Next, let's rewrite the numbers in the last column.

$n$	$(2 i)^{n}$	Location	Distance to the Origin
$0$	$1$	Real-axis	$2^{0}$
$1$	$2 i$	Imaginary-axis	$2^{1}$
$2$	$- 4$	Real-axis	$2^{2}$
$3$	$- 8 i$	Imaginary-axis	$2^{3}$
$4$	$(2 i)^{4}$	Real-axis	$?$
$5$	$(2 i)^{5}$	Imaginary-axis	$?$

With this last table we can say that the distance from $(2 i)^{4}$ to the origin is $2^{4} = 16 .$ Since it lies on the real-axis, its coordinates are $(16, 0) .$ Consequently, the distance from $(2 i)^{5}$ to the origin is $2^{5} = 32,$ and because it lies on the imaginary-axis its coordinates are $(0, 32) .$

$n$	$(2 i)^{n}$	Location	Distance to the Origin
$0$	$1$	Real-axis	$2^{0}$
$1$	$2 i$	Imaginary-axis	$2^{1}$
$2$	$- 4$	Real-axis	$2^{2}$
$3$	$- 8 i$	Imaginary-axis	$2^{3}$
$4$	$16$	Real-axis	$16$
$5$	$32 i$	Imaginary-axis	$32$

Exercise