Math Problem Statement
Simplify: 5i^22
Solution
To simplify , we need to analyze the powers of , where is the imaginary unit defined as . The powers of follow a repeating cycle every 4 powers:
i^1 &= i, \\ i^2 &= -1, \\ i^3 &= -i, \\ i^4 &= 1, \\ i^5 &= i, \text{ and so on.} \end{aligned}$$ ### Step 1: Find the remainder of $$22 \div 4$$ To determine where $$i^{22}$$ falls in the cycle, divide 22 by 4: $$22 \div 4 = 5 \text{ remainder } 2.$$ Thus: $$i^{22} = i^{4 \cdot 5 + 2} = (i^4)^5 \cdot i^2.$$ Since $$i^4 = 1$$, raising it to any power will still result in 1. Therefore: $$i^{22} = 1 \cdot i^2 = -1.$$ ### Step 2: Multiply by 5 Now, substitute $$i^{22} = -1$$ back into the expression: $$5i^{22} = 5 \cdot (-1) = -5.$$ ### Final Answer: The simplified form of $$5i^{22}$$ is: $$\boxed{-5}.$$ --- Would you like further details or clarification on any step? ### Related Questions: 1. How do powers of $$i$$ repeat, and why is the cycle every 4? 2. Simplify $$3i^{15}$$. 3. Find the value of $$i^{50}$$. 4. Explain why $$i^2 = -1$$ using the definition of $$i$$. 5. Simplify $$(2i)^6$$. ### Tip: When simplifying powers of $$i$$, always divide the exponent by 4 and use the remainder to determine the result based on the cycle $$i, -1, -i, 1$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Imaginary Numbers
Powers of i
Algebraic Simplification
Formulas
i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1 (Powers of i Cycle)
Theorems
Repetition of Powers of i every 4 cycles
Suitable Grade Level
Grades 9-11