Simplifying Powers of i - dummies

By Mary Jane Sterling

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, i3,003, i54,321, and i111,002

    i444 = 1; i3,003 = –i; i54,321 = i; i111,002 = –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, i3,003 = i4(750)+3 = –i, i54,321 = i4(13,580)+1 = i, and i111,002 = i4(27,750)+2 = –1.

Practice questions

  1. Simplify: i45

  2. Simplify: i60

  3. Simplify: i4,007

  4. Simplify: i2,002

Following are answers to the practice questions:

  1. The answer is i.

    Rewrite the term as i4(11)+1 = i.

  2. The answer is 1.

    Rewrite the term as i4(15) = 1.

  3. The answer is –i.

    Rewrite the term as i4(1,001)+3 = –i.

  4. The answer is –1.

    Rewrite the term as i4(500)+2 = –1.