1. **State the problem:** Simplify the expression $\frac{1}{i^{50}}$ where $i$ is the imaginary unit with the property $i^2 = -1$. 2. **Recall the powers of $i$:** The powers of $i$ cycle every 4 steps: $$i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1, \quad \text{and then it repeats}.$$ 3. **Reduce the exponent modulo 4:** Since powers of $i$ repeat every 4, find $50 \mod 4$: $$50 \div 4 = 12 \text{ remainder } 2,$$ so $$i^{50} = i^{4 \times 12 + 2} = (i^4)^{12} \times i^2 = 1^{12} \times (-1) = -1.$$ 4. **Substitute back:** $$\frac{1}{i^{50}} = \frac{1}{-1} = -1.$$ **Final answer:** $$\boxed{-1}$$