NZ Level 8 (NZC) Level 3 (NCEA) [In development] 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×i4 $=$= $16\times i^2\times i^2$16×i2×i2 $=$= $16\times\left(-1\right)\times\left(-1\right)$16×(−1)×(−1) $=$= $16$16

##### Example 2

Simplify $4i^2-3i^3-7i^4$4i23i37i4

 $4i^2-3i^3-7i^4$4i2−3i3−7i4 $=$= $4\times-1-3\times-i-7\times1$4×−1−3×−i−7×1 $=$= $-4--3i-7$−4−−3i−7 $=$= $3i-11$3i−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

#### M8-9

Manipulate complex numbers and present them graphically

#### 91577

Apply the algebra of complex numbers in solving problems