Math Problem Statement

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

Solution

We can solve i36i^{36} by using the properties of powers of ii, where ii is the imaginary unit. The powers of ii repeat in cycles of 4 as follows:

i1=i,i2=1,i3=i,i4=1i^1 = i,\quad i^2 = -1,\quad i^3 = -i,\quad i^4 = 1

Thus, for any integer nn, the powers of ii repeat every 4 steps. Therefore, to simplify i36i^{36}, we find the remainder when 36 is divided by 4:

36÷4=9 remainder 036 \div 4 = 9 \text{ remainder } 0

Since the remainder is 0, i36i^{36} is equivalent to i0i^0, which equals 1.

Therefore, i36=1i^{36} = 1.

Would you like more details or have any questions?

Here are 5 related questions:

  1. What is i50i^{50}?
  2. How do you simplify i99i^{99}?
  3. What is the value of i123i^{123}?
  4. What is the general form for ini^n for any integer nn?
  5. How do powers of ii relate to polar coordinates?

Tip: To simplify powers of ii, always reduce the exponent modulo 4, as the pattern repeats every 4 powers.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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