Powers of i Calculator
Use our powers of i calculator to calculate power of i instantly (iⁿ). Learn how to calculate powers in math, how to calculate power using the repeating i pattern, and common questions like can you add powers and what happens when you add powers.
What are Powers of i?
The imaginary unit i is defined by i² = −1. When you raise i to different powers, the results repeat in a simple cycle of four values.
That repeating cycle makes it easy to calculate power of i (iⁿ) without doing long multiplication.
This powers of i calculator finds iⁿ for any integer exponent n.
Powers of i Pattern and Formula
Powers of i repeat every 4. You can compute iⁿ by reducing n modulo 4.
After this, the pattern repeats: i^4 = 1, i^5 = i, and so on.
Use the remainder when dividing n by 4 to pick the result from the cycle.
Reduce the exponent by mod 4, then use the cycle.
Any exponent divisible by 4 gives 1.
How to Use the Powers of i Calculator
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Enter the exponent n.
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The calculator finds r = n mod 4.
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It returns the matching value from the cycle (1, i, −1, −i).
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If n is blank, you’ll see a default result like iⁿ = ?.
Frequently Asked Questions
Use the repeating pattern: i^0=1, i^1=i, i^2=−1, i^3=−i, then repeat every 4. Compute n mod 4 to pick the result.
It’s a tool that computes i^n quickly by using the mod 4 cycle.
Powers represent repeated multiplication: a^n means multiply a by itself n times (for integer n). Patterns and rules (like exponent laws) often simplify the work.
Most calculators have a power key (often x^y). For i^n specifically, many calculators rely on complex mode, but the mod 4 pattern is usually faster.
You generally cannot combine exponents when adding: a^m + a^n does not equal a^(m+n). Exponent rules like a^m · a^n = a^(m+n) apply to multiplication, not addition.
Powers of 2 are a different pattern (2^n). This calculator is specifically for powers of i, which repeat every 4.