Understanding Powers of the Imaginary Unit

The video tutorial explains how to simplify powers of the imaginary unit i. It uses the fact that i squared equals -1 to simplify expressions of i raised to various powers. The tutorial covers specific examples, including i to the 5th, 12th, 63rd, and 54th powers, demonstrating how to determine the number of i squared factors and the remainder. The results show that i raised to an even power simplifies to either 1 or -1, while i raised to an odd power simplifies to either i or -i. The video concludes with a summary of these four possible outcomes.

What is the value of i squared?

• -i

• 1

• i

• -1

When simplifying i to the 5th power, what is the remainder when 5 is divided by 2?

• 0

• 1

• 3

• 2

What is the simplified form of i to the 5th power?

• -i

• i

• 1

• -1

How many factors of i squared are in i to the 12th power?

• 5

• 6

• 8

• 7

What is the simplified result of i to the 12th power?

• -i

• 1

• -1

• i

What is the remainder when 63 is divided by 2?

• 3

• 2

• 1

• 0

What is the simplified form of i to the 63rd power?

• i

• -1

• 1

• -i

How many factors of i squared are in i to the 54th power?

• 26

• 27

• 28

• 29

What is the simplified result of i to the 54th power?

• 1

• -1

• i

• -i

When i is raised to an even power, what are the possible results?

• 1 or -1

• i or -i

• -1 or i

• 0 or 1