Powers of i

Introduction to Powers of i

The imaginary unit i is defined as the square root of -1, that is, . In precalculus and complex number algebra, it is important to understand how to simplify expressions involving powers of i. All exponent rules apply to powers of i, and any power of i can be reduced to one of four possible values.

• Key Point 1: The powers of i repeat in a cycle of four.

• Key Point 2: Any power of i can be simplified to either 1, -1, i, or -i.

Table: Powers of i

The following table summarizes the first few powers of i and their results:

How to Evaluate Higher Powers of i

To simplify higher powers of i, express the exponent as a multiple of 4 plus a remainder. The value of depends on the remainder when n is divided by 4:

• If , then

• If , then

• If , then

• If , then

Example: Simplifying Powers of i

• Example 1: Simplify

  • Divide 23 by 4: remainder 3

  • So,

• Example 2: Simplify

  • Divide 100 by 4: remainder 0

  • So,

Shortcut for Very High Powers

For very large exponents, use the remainder after dividing by 4 to quickly determine the value:

Practice Problems

• Problem 1: Simplify

  • remainder 0

  • Answer:

• Problem 2: Simplify

  • remainder 3

  • Answer:

Summary: Powers of i cycle every four exponents. To simplify any power of i, divide the exponent by 4 and use the remainder to determine the result using the table above.