Class XI

Powers of i

Lesson

Activity

Before we embark on further explorations with our complex number $i$i, let's have a look at what happens when we take successive powers of $i$i.

Try this activity yourself first, before checking out my solution.

Make a list of powers of $i$i, up to $i^{15}$i15.

Simplify the results.

Then generalise the pattern.

(see here for the solution)

Now that you have explored how to simplify powers of $i$i, the only other thing to do is combine this with other algebraic simplifications.

Here are some examples

Example 1

Simplify $(2i)^4$(2i)4

We need to remember our index laws here, so $\left(ab\right)^n=a^n\times b^n$(ab)n=an×bn

Thus,

$\left(2\times1\right)^4$(2×1)4	$=$=	$2^4\times i^4$24×`i`4
	$=$=	$16\times i^2\times i^2$16×`i`2×`i`2
	$=$=	$16\times\left(-1\right)\times\left(-1\right)$16×(−1)×(−1)
	$=$=	$16$16

Example 2

Simplify $4i^2-3i^3-7i^4$4i2−3i3−7i4

$4i^2-3i^3-7i^4$4`i`2−3`i`3−7`i`4	$=$=	$4\times-1-3\times-i-7\times1$4×−1−3×−`i`−7×1
	$=$=	$-4--3i-7$−4−−3`i`−7
	$=$=	$3i-11$3`i`−11

More Worked Examples

QUESTION 1

Simplify $2i^7$2i7.

QUESTION 2

Simplify $\left(2i\right)^9$(2i)9.

QUESTION 3

Simplify $\left(\sqrt{5}i\right)^6$(√5i)6.

Outcomes

11.A.CNQE.1

Need for complex numbers, especially √-1, to be motivated by inability to solve every quadratic equation. Brief description of algebraic properties of complex numbers. Argand plane and polar representation of complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations in the complex number system.