Powers of i Calculator

The imaginary unit, denoted as i, is defined by the fundamental property that its square is negative one: i² = -1. When you raise i to higher integer powers, the resulting values follow a predictable, repeating cycle of four terms: i¹ = i, i² = -1, i³ = -i, and i⁴ = 1. For any integer exponent n, the value of iⁿ depends entirely on the remainder of n divided by 4. This relationship is mathematically expressed using the modulo operation: remainder2 = |n| % 4. If the remainder is 0, the result is 1; if 1, the result is i; if 2, the result is -1; and if 3, the result is -i.

Let us evaluate i raised to the power of 42 (where n = 42). First, we take the absolute value of the exponent, which is 42. Next, we divide 42 by 4 to find the remainder: 42 ÷ 4 = 10 with a remainder of 2. Using our formula, remainder2 = 42 % 4 = 2. Looking at our cyclic pattern, a remainder of 2 corresponds directly to i². Therefore, i⁴² simplifies neatly to -1.

When solving expressions manually, always divide the exponent by 4 and focus solely on the remainder rather than trying to multiply i repeatedly. For negative exponents, remember the reciprocal rule: i^-n is equivalent to 1 / iⁿ, which can also be simplified using the standard four-step cycle. Watch out for large exponents in polynomial equations; breaking them down using exponent rules (like i^{a + b} = i^a · i^b) makes complex number arithmetic much easier to manage.