Final answer:
The equivalent imaginary solution to i^34 is -1. This is determined by realizing that the powers of i are periodic with a period of 4, and i raised to the power of 34 results in -1 after simplification.
Step-by-step explanation:
The question is asking for the equivalent imaginary solution to the expression ‖i^34‖. Using the properties of imaginary numbers, specifically the powers of ‖i‖ (the imaginary unit), we know that:
- ‖i‖ to the power of 1 is ‖i‖
- ‖i‖ to the power of 2 is -1
- ‖i‖ to the power of 3 is -‖i‖
- ‖i‖ to the power of 4 is 1
Every fourth power of ‖i‖ loops back to 1, so powers of ‖i‖ are periodic with a period of 4. Thus, we can simplify ‖i^34‖ by dividing 34 by 4, which gives us 8 with a remainder of 2. Therefore:
‖i^34 = i^(4×8+2) = (i^4)^8 × i^2 = 1^8 × (-1) = -1‖
The equivalent imaginary solution to ‖i^34‖ is -1.