1. **State the problem:** We need to simplify the expression $$M=\left[(1+\frac{6}{1})(1+\frac{7}{1})(1+\frac{8}{1})(1-\frac{6}{1})(1-\frac{7}{1})(1-\frac{8}{1})\right]^{-1}$$. 2. **Rewrite the terms:** Since the denominators are 1, simplify inside the parentheses: $$M=\left[(1+6)(1+7)(1+8)(1-6)(1-7)(1-8)\right]^{-1}$$ 3. **Calculate each term:** $$(7)(8)(9)(-5)(-6)(-7)$$ 4. **Multiply the positive terms:** $$7 \times 8 = 56$$ $$56 \times 9 = 504$$ 5. **Multiply the negative terms:** $$(-5) \times (-6) = 30$$ $$30 \times (-7) = -210$$ 6. **Multiply all terms together:** $$504 \times (-210) = -105840$$ 7. **Apply the exponent -1 (reciprocal):** $$M = \frac{1}{-105840} = -\frac{1}{105840}$$ **Final answer:** $$M = -\frac{1}{105840}$$