Which of the following is equivalent to the complex number 
\[ i^{36}\]?

We can solve \( i^{36} \) by using the properties of powers of \( i \), where \( i \) is the imaginary unit. The powers of \( i \) repeat in cycles of 4 as follows:

\[
i^1 = i,\quad i^2 = -1,\quad i^3 = -i,\quad i^4 = 1
\]

Thus, for any integer \( n \), the powers of \( i \) repeat every 4 steps. Therefore, to simplify \( i^{36} \), we find the remainder when 36 is divided by 4:

\[
36 \div 4 = 9 \text{ remainder } 0
\]

Since the remainder is 0, \( i^{36} \) is equivalent to \( i^0 \), which equals 1.

Therefore, \( i^{36} = 1 \).

Would you like more details or have any questions?

Here are 5 related questions:
1. What is \( i^{50} \)?
2. How do you simplify \( i^{99} \)?
3. What is the value of \( i^{123} \)?
4. What is the general form for \( i^n \) for any integer \( n \)?
5. How do powers of \( i \) relate to polar coordinates?

**Tip:** To simplify powers of \( i \), always reduce the exponent modulo 4, as the pattern repeats every 4 powers.

Which of the following is equivalent to the complex number i^{36}?

Learn how to simplify i^{36} using the properties of complex numbers and modular arithmetic. This detailed explanation shows how the powers of i repeat in cycles of 4, helping students understand how to solve problems involving powers of imaginary numbers. Suitable for students in Grades 10-12.

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