1. **State the problem:** Simplify the expression $$(-2)^2 - (-2)(+3) + (-2)^7 : (-2)^4 - [(-2)^8 \times (-2)^2] : (-2)$$ where ":" denotes division. 2. **Recall the rules:** - Powers of negative numbers: $$(-a)^n = (-1)^n a^n$$ - Division of powers with the same base: $$\frac{a^m}{a^n} = a^{m-n}$$ - Multiplication of powers with the same base: $$a^m \times a^n = a^{m+n}$$ 3. **Calculate each power:** - $$(-2)^2 = (-1)^2 \times 2^2 = 1 \times 4 = 4$$ - $$(-2)^7 = (-1)^7 \times 2^7 = -1 \times 128 = -128$$ - $$(-2)^4 = (-1)^4 \times 2^4 = 1 \times 16 = 16$$ - $$(-2)^8 = (-1)^8 \times 2^8 = 1 \times 256 = 256$$ - $$(-2)^2 = 4$$ (already calculated) 4. **Rewrite the expression with values:** $$4 - (-2)(3) + \frac{-128}{16} - \left[256 \times 4\right] : (-2)$$ 5. **Simplify multiplication and division step-by-step:** - $$- (-2)(3) = +6$$ - $$\frac{-128}{16} = -8$$ - $$256 \times 4 = 1024$$ - Division by $$(-2)$$: $$\frac{1024}{-2}$$ 6. **Apply division with cancellation:** $$\frac{1024}{-2} = \frac{\cancel{1024}}{\cancel{-2}} = -512$$ 7. **Substitute back:** $$4 + 6 - 8 - (-512)$$ 8. **Simplify addition and subtraction:** $$4 + 6 = 10$$ $$10 - 8 = 2$$ $$2 - (-512) = 2 + 512 = 514$$ **Final answer:** $$514$$