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 i

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 i

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 i

^{2}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 i^{4}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 i

^{4}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 i^{4}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 i

^{148}with 1 and i^{3}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 i^{2}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 i^{4}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. $$i^{59}$$

Please choose the best answer.

A

$$i$$

B

$$1$$

C

$$-1$$

D

$$59$$

E

$$-i$$

Example #2

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

Please choose the best answer.

A

$$i$$

B

$$-1$$

C

$$1$$

D

$$-i$$

E

$$\frac{i}{73}$$

Example #3

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

Please choose the best answer.

A

$$-214$$

B

$$\frac{-i}{214}$$

C

$$i$$

D

$$-1$$

E

$$-i$$

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