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 i^{2} = –1. If you want i^{3}, you compute it by writing i^{3} = i^{2} x i = –1 x i = –i. Also, i^{4} = i^{2} x i^{2} = (–1)(–1) = 1.
And then the values of the powers start repeating themselves, because i^{5} = i, i^{6} = –1, i^{7} = –i, and i^{8} = 1. So, what do you do if you want a higher power, such as i^{345}, or something else pretty high up there?
You don’t want to have to write out all the powers up to i^{345} using the pattern (not when you could be whitewater 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:

i^{4}^{n} = 1

i^{4}^{n}^{+1} = i

i^{4}^{n}^{+2} = –1

i^{4}^{n}^{+3} = –i
Sample question

Simplify the following: i^{444}, i^{3,003}, i^{54,321}, and i^{111,002}
i^{444} = 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 i^{444} = i^{4(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

Simplify: i^{45}

Simplify: i^{60}

Simplify: i^{4,007}

Simplify: i^{2,002}
Following are answers to the practice questions:

The answer is i.
Rewrite the term as i^{4(11)+1} = i.

The answer is 1.
Rewrite the term as i^{4(15)} = 1.

The answer is –i.
Rewrite the term as i^{4(1,001)+3} = –i.

The answer is –1.
Rewrite the term as i^{4(500)+2} = –1.