1. The problem is to find the value of $i^{45}$, where $i$ is the imaginary unit with the property $i^2 = -1$. 2. Important rule: Powers of $i$ cycle every 4 steps because: $$i^1 = i$$ $$i^2 = -1$$ $$i^3 = -i$$ $$i^4 = 1$$ Then the cycle repeats. 3. To simplify $i^{45}$, find the remainder when 45 is divided by 4: $$45 \div 4 = 11 \text{ remainder } 1$$ 4. Using the cycle, $i^{45} = i^{4 \times 11 + 1} = (i^4)^{11} \times i^1 = 1^{11} \times i = i$. 5. Therefore, the value of $i^{45}$ is $i$.