Simplify i raised to the power 27. What is the result?

1. The powers of i repeat in a cycle of length 4, running i, negative 1, negative i, then 1. 2. So a power of i depends only on the remainder when the exponent is divided by 4. 3. Divide 27 by 4 to get 6 with a remainder of 3. 4. That means i to the 27 equals i to the 3. 5. The third entry of the cycle is negative i, so the answer is negative i. 6. Answering positive i would use a remainder of 1, which comes from dividing 25 rather than 27. 7. Answering negative one would use a remainder of 2. 8. Answering positive one would use a remainder of 0, which happens only for multiples of 4. _Source: OpenStax Precalculus (CC BY 4.0), section 3.1 Complex Numbers_