Share
Simplify i^6 + i^16 + i^26 + i^36 + i^46 + i^56
Published on August 2, 2024
Algebra
This problem simplifies a complex expression involving powers of the imaginary unit 'i'. The expression, i^6 + i^16 + i^26 + i^36 + i^46 + i^56, surprisingly evaluates to 0.
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:
-
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.
-
Simplify using the cyclical pattern:
-
-
Substitute and calculate:Our expression becomes:
(-1) + 1 + (-1) + 1 + (-1) + 1 = 0
Therefore, the simplified form of the expression is 0.