# Evaluate Powers Of I Worksheet

## 6 problems

Evaluating powers of $$i$$ involves understanding the cyclic nature of the imaginary unit $$i$$, where $$i^1 = i$$, $$i^2 = -1$$, $$i^3 = -i$$, and $$i^4 = 1$$. This pattern repeats every four powers due to $$i$$'s property $$i^4 = 1$$. Evaluate powers of $$i$$ worksheet helps practice these calculations, useful in algebra for simplifying complex expressions and solving equations involving $$i$$. Understanding these powers is fundamental in manipulating imaginary numbers effectively in mathematical contexts.

Algebra 2
Real And Complex Numbers

## How Will This Worksheet on "Evaluate Powers of i" Benefit Your Student's Learning?

• Helps students understand imaginary numbers and their role in complex numbers.
• Simplifies algebraic expressions involving imaginary units, crucial for advanced math classes.
• Recognizing the cyclic pattern of $$i$$ improves pattern recognition in math.
• Solving equations with $$i$$ powers requires logical thinking and using math rules effectively to find solutions.
• Builds a foundation for advanced topics in complex analysis, including exponentials and identities.

## How to Evaluate Powers of i?

• $$i$$ follows a cyclic pattern: $$i^1 = i$$, $$i^2 = -1$$, $$i^3 = -i$$, $$i^4 = 1$$, and then it repeats.
• To find $$i^n$$, divide $$n$$ by 4 to determine the remainder $$r$$. Use the corresponding $$i^r$$ from the cycle (1 for $$r = 0$$, $$i$$ for $$r = 1$$, $$-1$$ for $$r = 2$$, $$-i$$ for $$r = 3$$).
• Practice calculating powers of $$i$$ using worksheets to reinforce understanding.

## Solved Example

Q. What is the value of $i^4$?
Solution:
1. Is Exponent Even or Odd?: $i^4$ Is exponent even or odd? Exponent: $4$$\newline$ $4$ is an even number.
2. Convert to $i^2$ Base: Convert $i^4$ into an expression with $i^2$ as the base. $\newline$$i^4$ $=$ $i^{2\times 2}$ $=$ $(i^2)^2$
3. Evaluate $(i^2)^2$: $\newline$ $i^2 = -1$$\newline$ $(i^2)^2$ $= (-1)^2$ $= 1$

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