Simplifying Powers of i

Performing operations on complex numbers requires multiplying by i and simplifying powers of i. By definition, i = the square root of –1, so i² = –1. If you want i³, you compute it by writing i³ = i² x i = –1 x i = –i. Also, i⁴ = i² x i² = (–1)(–1) = 1.

And then the values of the powers start repeating themselves, because i⁵ = i, i⁶ = –1, i⁷ = –i, and i⁸ = 1. So, what do you do if you want a higher power, such as i³⁴⁵, or something else pretty high up there?

You don’t want to have to write out all the powers up to i³⁴⁵ using the pattern (not when you could be white-water rafting or cleaning your room or watching the Cubs win the World Series!). Instead, use the following rule.

To compute the value of a power of i, determine whether the power is a multiple of 4, one more than a multiple of 4, two more than, or three more than a multiple of 4. Then apply the following:

• i⁴ⁿ = 1

• i⁴ⁿ⁺¹ = i

• i⁴ⁿ⁺² = –1

• i⁴ⁿ⁺³ = –i

Sample question

1. Simplify the following: i⁴⁴⁴, i^3,003, i^54,321, and i^111,002

   i⁴⁴⁴ = 1; i^3,003 = –i; i^54,321 = i; i^111,002 = –1. Writing the power of i as a multiple of 4 and what’s left over (you know, the remainder), you get i⁴⁴⁴ = i⁴⁽¹¹¹⁾ = 1, i^3,003 = i^4(750)+3 = –i, i^54,321 = i^4(13,580)+1 = i, and i^111,002 = i^4(27,750)+2 = –1.

Practice questions

1. Simplify: i⁴⁵

2. Simplify: i⁶⁰

3. Simplify: i^4,007

4. Simplify: i^2,002

Following are answers to the practice questions:

1. The answer is i.

   Rewrite the term as i^4(11)+1 = i.

2. The answer is 1.

   Rewrite the term as i⁴⁽¹⁵⁾ = 1.

3. The answer is –i.

   Rewrite the term as i^4(1,001)+3 = –i.

4. The answer is –1.

   Rewrite the term as i^4(500)+2 = –1.