Lesson Objectives
• Learn how to simplify powers of the imaginary unit i
• Learn how to rationalize a denominator with i

## How to Simplify Powers of i

We may be asked to simplify powers of i. We will use the fact that i2 is (-1) along with the rules for exponents to simplify powers of i. We should note the first few powers of i: $$i^0=1$$ $$i^1=i$$ $$i^2=-1$$ $$i^3=i^2 \cdot i=-1 \cdot i=-i$$ $$i^4=i^2 \cdot i^2=-1 \cdot -1=1$$ If we are trying to simplify and our whole-number exponent on i is 4 or less, we can use the rules above. Alternatively, what we want to do is think about the next number going down that is divisible by 4. We will use the fact that i4 is 1, and 1 raised to any power is still 1. Let's take a look at a few examples.
Example 1: Simplify each. $$i^{37}$$ Since 36 is divisible by 4, we will use the product rule for exponents to rewrite our problem as: $$i^{37}=i^{36}\cdot i$$ Now, we can use our power to power rule to rewrite the problem as: $$(i^4)^9 \cdot i$$ We know that i4 is 1, we can replace this in our problem: $$(i^4)^9 \cdot i=1^9 \cdot i=1 \cdot i=i$$ Example 2: Simplify each. $$i^{94}$$ Since 92 is divisible by 4, we will use the product rule for exponents to rewrite our problem as: $$i^{92}\cdot i^2$$ Now, we can use our power to power rule to rewrite the problem as: $$(i^4)^{23}\cdot i^2$$ We know that i4 is 1, we can replace this in our problem: $$(i^4)^{23}\cdot i^2=1^{23}\cdot i^2=i^2=-1$$

### Rationalizing a Denominator with i

In some cases, we may end up with i in our denominator. When this happens, we technically need to rationalize the denominator since i is the square root of -1, and radicals are not allowed in the denominator when simplifying. Let's look at an example.
Example 3: Simplify each. $$i^{-151}$$ First, let's use our rule for negative exponents: $$\frac{1}{i^{151}}$$ Now, we know that 148 is divisible by 4, let's use the product rule for exponents to rewrite our problem as: $$\frac{1}{i^{148}\cdot i^{3}}$$ $${i^{148}}=(i^{4})^{37}=1^{37}=1$$ $${i^3=-i}$$ We can replace i148 with 1 and i3 with -i: $$\frac{1}{1 \cdot -i}$$ $$-\frac{1}{i}$$ Now, we can rationalize the denominator by multiplying by i/i: $$-\frac{1}{i}\cdot \frac{i}{i}=-\frac{i}{i^2}$$ We know that i2 is -1: $$-\frac{i}{i^2}=-\frac{i}{-1}=i$$

### An Alternative Approach

Additionally, we could have just multiplied by 1 in the form of i4 to create the least positive exponent on i. $$i^{-151}$$ Think about positive numbers going up from 151 that are divisible by 4. The next one would be 152. $$i^{152}=(i^{4})^{38}=1$$ We can always multiply by 1 and leave a number unchanged: $$i^{-151}\cdot i^{152}=i$$ We get the same answer with much less effort.

#### Skills Check:

Example #1

Simplify each. $$i^{59}$$

A
$$i$$
B
$$1$$
C
$$-1$$
D
$$59$$
E
$$-i$$

Example #2

Simplify each. $$i^{-73}$$

A
$$i$$
B
$$-1$$
C
$$1$$
D
$$-i$$
E
$$\frac{i}{73}$$

Example #3

Simplify each. $$\frac{1}{i^{214}}$$

A
$$-214$$
B
$$\frac{-i}{214}$$
C
$$i$$
D
$$-1$$
E
$$-i$$