Simplify i^6 + i^16 + i^26 + i^36 + i^46 + i^56

Published on August 2, 2024

Question

simplify it

Answer

Let’s simplify the expression:
Key Idea: The imaginary unit ‘i’ follows a cyclical pattern:
Notice that the powers of ‘i’ cycle through these four values.
Steps:
  1. Divide the exponents by 4: Focus on the remainder after dividing each exponent by 4. This is because every power of equals 1.
    • 6 divided by 4 leaves a remainder of 2.
    • 16 divided by 4 leaves a remainder of 0.
    • 26 divided by 4 leaves a remainder of 2.
    • 36 divided by 4 leaves a remainder of 0.
    • 46 divided by 4 leaves a remainder of 2.
    • 56 divided by 4 leaves a remainder of 0.
  2. Simplify using the cyclical pattern:
  3. Substitute and calculate:
    Our expression becomes:
    (-1) + 1 + (-1) + 1 + (-1) + 1 = 0
Therefore, the simplified form of the expression is 0.
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