1. **State the problem:** Simplify the expression $ (1-2x)^3 $. 2. **Formula used:** The cube of a binomial $ (a-b)^3 $ is expanded using the formula: $$ (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 $$ 3. **Identify terms:** Here, $ a = 1 $ and $ b = 2x $. 4. **Apply the formula:** $$ (1-2x)^3 = 1^3 - 3 \times 1^2 \times (2x) + 3 \times 1 \times (2x)^2 - (2x)^3 $$ 5. **Calculate each term:** - $1^3 = 1$ - $3 \times 1^2 \times 2x = 6x$ - $(2x)^2 = 4x^2$, so $3 \times 1 \times 4x^2 = 12x^2$ - $(2x)^3 = 8x^3$ 6. **Put it all together:** $$ (1-2x)^3 = 1 - 6x + 12x^2 - 8x^3 $$ 7. **Final answer:** $$ \boxed{1 - 6x + 12x^2 - 8x^3} $$