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Simplifying Powers Of I Calculator

Powers of i (imaginary unit):

\[ i = \sqrt{-1} \] \[ i^1 = i \] \[ i^2 = -1 \] \[ i^3 = -i \] \[ i^4 = 1 \] \[ \text{Pattern repeats every 4 powers} \]

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1. What Are Powers of i?

The imaginary unit i is defined as the square root of -1. When we raise i to different powers, we get results that cycle through a predictable pattern every 4 powers.

2. How the Calculator Works

The calculator uses the cyclical nature of powers of i:

\[ i^n = i^{(n \mod 4)} \]

Where:

Explanation: Any power of i can be simplified by finding the remainder when the exponent is divided by 4.

3. The Cyclic Pattern

Details: The powers of i follow a repeating cycle every 4 exponents: i → -1 → -i → 1 → i → -1 → -i → 1 → and so on.

4. Using the Calculator

Tips: Enter any non-negative integer power of i to see its simplified form. The calculator works for very large exponents by using the modulo operation.

5. Frequently Asked Questions (FAQ)

Q1: What is the imaginary unit i?
A: i is defined as the square root of -1, the fundamental unit of imaginary numbers.

Q2: Why do powers of i cycle every 4?
A: Because i^4 = (i^2)^2 = (-1)^2 = 1, which brings us back to the starting point.

Q3: How would you simplify i^100?
A: Since 100 mod 4 = 0, i^100 = i^0 = 1.

Q4: Can this calculator handle negative exponents?
A: The current version only accepts non-negative integers, but negative exponents can be handled by using i^-n = 1/i^n.

Q5: What are some applications of powers of i?
A: Powers of i are essential in complex number calculations, electrical engineering (phasors), quantum mechanics, and signal processing.

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