Math Problem Statement

Simplify i^{118}

Solution

To simplify i118i^{118}, where ii is the imaginary unit, let's recall the powers of ii, which cycle every 4 steps:

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

Thus, the powers of ii repeat every 4. To determine i118i^{118}, we divide 118118 by 44 and find the remainder:

118÷4=29remainder2118 \div 4 = 29 \quad \text{remainder} \, 2

This means:

i118=i2i^{118} = i^2

From the cycle, i2=1i^2 = -1. Therefore:

i118=1i^{118} = -1

Final Answer:

1\boxed{-1}

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Related Questions:

  1. Simplify i75i^{75}.
  2. What is the value of i200i^{200}?
  3. Find i123i^{123} in simplified form.
  4. Explain the powers of ii and their cycles.
  5. Simplify ini^{n} for any integer nn.

Tip:

The powers of ii 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