Math Problem Statement
Simplify i^{118}
Solution
To simplify , where is the imaginary unit, let's recall the powers of , which cycle every 4 steps:
Thus, the powers of repeat every 4. To determine , we divide by and find the remainder:
This means:
From the cycle, . Therefore:
Final Answer:
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Related Questions:
- Simplify .
- What is the value of ?
- Find in simplified form.
- Explain the powers of and their cycles.
- Simplify for any integer .
Tip:
The powers of repeat every 4 steps. Always divide the exponent by 4 and use the remainder to determine the simplified result.
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Powers of i
Formulas
Powers of i cycle every 4 steps: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1
Theorems
Cyclic nature of powers of the imaginary unit i
Suitable Grade Level
Grades 10-12