Math Problem Statement
simplifiy i138 Show all work including steps to solve
Solution
To simplify the expression , where is the imaginary unit, follow these steps:
Step 1: Recall the powers of
The powers of repeat every four terms:
i^1 & = i, \\ i^2 & = -1, \\ i^3 & = -i, \\ i^4 & = 1. \end{aligned}$$ So, the cycle repeats every four powers. ### Step 2: Determine the remainder when 138 is divided by 4 To find which part of the cycle $$i^{138}$$ corresponds to, divide 138 by 4 and find the remainder. $$138 \div 4 = 34 \text{ remainder } 2$$ This means: $$138 = 4 \times 34 + 2$$ So, $$i^{138}$$ has the same value as $$i^2$$. ### Step 3: Use the power of $$i^2$$ From the first step, we know: $$i^2 = -1$$ ### Final Answer: Thus, $$i^{138} = -1$$. Would you like more details or have any other questions? Here are some related questions: 1. How do you simplify $$i^{200}$$? 2. What are the properties of imaginary numbers? 3. How do you express complex numbers in polar form? 4. What is the geometric interpretation of multiplying by $$i$$? 5. How do powers of $$i$$ relate to rotations in the complex plane? **Tip:** When working with powers of $$i$$, always reduce the exponent modulo 4 to simplify the expression.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Powers of Imaginary Unit
Formulas
i^n for n = 1, 2, 3, 4
Modulo operation
Theorems
Properties of Imaginary Unit i
Suitable Grade Level
Grades 9-12