Math
Powers of i Calculator
Type an exponent and get i^n simplified to 1, i, -1, or -i, with the division by 4 written out so you see why.
Because i^4 = 1, the powers cycle every four steps, and negative exponents fit the same cycle (i^-1 = -i). Switch to the coefficient mode to simplify expressions like (2i)^5 or (i/3)^-2, where the coefficient is raised to the power separately and multiplied by the reduced power of i.
Any integer, positive, negative, or huge (up to 50 digits in i^n mode).
Try an example
Result
Simplified
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- n mod 4
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- Cycle position
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- Rewrite
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- Decimal form
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What the powers of i calculator solves
The imaginary unit i is defined by i^2 = -1. Multiplying by i again gives i^3 = -i, and once more gives i^4 = 1, back to the start. So the powers of i repeat in a cycle of four: 1, i, -1, -i, 1, i, -1, -i, … To simplify i^n you only need the remainder when n is divided by 4. This calculator shows that division and the cycle lookup, and extends the idea to expressions like (2i)^5.
| n mod 4 | i^n | Examples |
|---|---|---|
| 0 | 1 | i^0, i^4, i^8, i^100, i^-4 |
| 1 | i | i^1, i^5, i^9, i^101, i^-3 |
| 2 | -1 | i^2, i^6, i^10, i^102, i^-2 |
| 3 | -i | i^3, i^7, i^11, i^23, i^-1 |
Negative exponents
i^-1 = 1/i. Multiply top and bottom by i: i/(i^2) = i/(-1) = -i. So the cycle runs backwards for negative The small raised number saying how many times to multiply the base by itself.: i^-1 = -i, i^-2 = -1, i^-3 = i, i^-4 = 1. Using a non-negative remainder handles this automatically: -1 = 4 × (-1) + 3, remainder 3, and i^3 = -i.
Powers with a coefficient
For (c·i)^n, raise the The number multiplying a variable, like the 3 in 3x. and i separately: (2i)^5 = 2^5 × i^5 = 32 × i = 32i, and (2i)^6 = 64 × i^2 = -64. Fractions work the same way, (i/2)^3 = (1/8)(-i) = -i/8, and negative exponents invert the coefficient: (2i)^-2 = (1/4)(i^-2) = -1/4.
How to use it
- Enter the exponent n; read i^n and the n mod 4 step that produced it.
- Choose the coefficient mode and enter c to simplify (c·i)^n exactly.
- Open Show the work to see the cycle, the division by 4, and the coefficient power.
How to read the answer
The simplified value is always one of 1, i, -1, -i for a pure power. Remainder 0 or 2 gives a real number (1 or -1); remainder 1 or 3 gives a purely imaginary number (i or -i). With a coefficient the answer is c^n times that unit, so it is real when n mod 4 is 0 or 2 and imaginary otherwise. The decimal form writes the answer as a + bi.
Common mistakes and edge cases
- Using n mod 2 instead of n mod 4: i^6 = -1, not 1.
- Taking the remainder of a negative exponent as negative: -1 mod 4 is 3, so i^-1 = i^3 = -i.
- Forgetting to raise the coefficient: (2i)^5 is 32i, not 2i or 10i.
- Writing i^3 = i; the correct value is -i because i^3 = i^2 × i = -1 × i.
- Treating i^0 as 0; any non-zero number to the power 0 is 1.
Worked examples
i^23
23 mod 4 = 3 → -i.
Simplified
-i
i^100
Multiple of 4 → 1.
Simplified
1
i^0
Anything to the 0 is 1.
Simplified
1
i^-1
1/i = -i.
Simplified
-i
Huge exponent
i^(10^20 + 2) = -1.
Simplified
-1
(2i)^5
32 × i = 32i.
Simplified
32i
(2i)^6
64 × (-1) = -64.
Simplified
-64
(i/2)^3
(1/8) × (-i) = -i/8.
Simplified
-1i/8
(-3i)^2
9 × (-1) = -9.
Simplified
-9
Fractional exponent
Not supported.
Simplified
Error
Frequently asked questions
What is i to the power of 2?+
i^2 = -1 by definition of the imaginary unit.
What are the powers of i?+
They cycle: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, and then repeat. i^n depends only on n mod 4.
How do I simplify i^23?+
Divide 23 by 4: remainder 3. So i^23 = i^3 = -i.
What is i to a negative power?+
i^-1 = -i, i^-2 = -1, i^-3 = i, i^-4 = 1. Use the non-negative remainder of n mod 4: -1 mod 4 = 3, giving i^3 = -i.
What is (2i)^5?+
2^5 × i^5 = 32 × i = 32i.
Is i^n ever a real number?+
Yes, when n mod 4 is 0 or 2: i^n is 1 or -1. Otherwise it is i or -i, purely imaginary.
What is i^0?+
1, like any non-zero base raised to the power 0.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026