1. **State the problem:** Simplify the expression $ (2x)^3 - (2y)^3 $. 2. **Recall the formula:** This is a difference of cubes, which follows the identity $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$ where $a = 2x$ and $b = 2y$. 3. **Apply the formula:** Substitute $a$ and $b$ into the formula: $$ (2x)^3 - (2y)^3 = (2x - 2y)((2x)^2 + (2x)(2y) + (2y)^2) $$ 4. **Calculate each term inside the second parentheses:** $$ (2x)^2 = 4x^2 $$ $$ (2x)(2y) = 4xy $$ $$ (2y)^2 = 4y^2 $$ 5. **Rewrite the expression:** $$ (2x - 2y)(4x^2 + 4xy + 4y^2) $$ 6. **Factor out the common factor 2 from the first parentheses:** $$ \cancel{2}(x - y) \times 4(x^2 + xy + y^2) $$ 7. **Simplify the constants:** $$ 2 \times 4 = 8 $$ 8. **Final simplified expression:** $$ 8(x - y)(x^2 + xy + y^2) $$