1. The problem is to evaluate the expression $i^{-23}$ where $i$ is the imaginary unit with the property $i^2 = -1$. 2. Recall the powers of $i$ cycle every 4 steps: $$i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1$$ 3. To simplify $i^{-23}$, first rewrite the negative exponent: $$i^{-23} = \frac{1}{i^{23}}$$ 4. Next, find $i^{23}$ by reducing the exponent modulo 4: $$23 \div 4 = 5 \text{ remainder } 3$$ So, $$i^{23} = i^{4 \times 5 + 3} = (i^4)^5 \times i^3 = 1^5 \times i^3 = i^3$$ 5. From the cycle, $i^3 = -i$, so: $$i^{23} = -i$$ 6. Substitute back: $$i^{-23} = \frac{1}{-i} = -\frac{1}{i}$$ 7. To simplify $\frac{1}{i}$, multiply numerator and denominator by $i$: $$-\frac{1}{i} = -\frac{1}{i} \times \frac{i}{i} = -\frac{i}{i^2}$$ 8. Since $i^2 = -1$, this becomes: $$-\frac{i}{-1} = -i \times (-1) = i$$ **Final answer:** $$i^{-23} = i$$