This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — To evaluate i^1265, we use the cyclic property of powers of i. The powers of i repeat in a cycle of 4: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. Step 1: Divide the exponent by 4 to find the remainder. We need to find the remainder of 1265 ÷ 4. 1265 ÷ 4 = 316 with a remainder of 1 This can be written as 1265 = 4 × 316 + 1. Step 2: Use the remainder to evaluate the expression. We can rewrite i^1265 using the remainder: i^1265 = i^4 × 316 + 1 Using the exponent rule a^m+n = a^m · a^n and (a^m)^n = a^mn: i^1265 = (i^4)^316 · i^1 Since i^4 = 1: i^1265 = (1)^316 · i i^1265 = 1 · i i^1265 = i Step 3: Compare the result with the given options. The calculated value is i, which corresponds to option A. The final answer is A. What's next?