Math Problem Statement
Which of the following is equivalent to the complex number [ i^{36}]?
Solution
We can solve by using the properties of powers of , where is the imaginary unit. The powers of repeat in cycles of 4 as follows:
Thus, for any integer , the powers of repeat every 4 steps. Therefore, to simplify , we find the remainder when 36 is divided by 4:
Since the remainder is 0, is equivalent to , which equals 1.
Therefore, .
Would you like more details or have any questions?
Here are 5 related questions:
- What is ?
- How do you simplify ?
- What is the value of ?
- What is the general form for for any integer ?
- How do powers of relate to polar coordinates?
Tip: To simplify powers of , always reduce the exponent modulo 4, as the pattern repeats every 4 powers.
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Powers of Imaginary Unit
Modular Arithmetic
Formulas
i^1 = i
i^2 = -1
i^3 = -i
i^4 = 1
i^n = i^{n mod 4}
Theorems
Powers of i repeat in cycles of 4
Suitable Grade Level
Grades 10-12