1. **State the problem:** Solve for $a$ in the equation $$\left(\frac{1}{9}\right)^{a+1} = 81^{8+a+1} \cdot 27^{2 - a}.$$\n\n2. **Rewrite bases as powers of 3:**\n- $9 = 3^2$, so $\frac{1}{9} = 3^{-2}$.\n- $81 = 3^4$.\n- $27 = 3^3$.\n\n3. **Rewrite the equation using these bases:**\n$$\left(3^{-2}\right)^{a+1} = \left(3^4\right)^{8+a+1} \cdot \left(3^3\right)^{2 - a}.$$\n\n4. **Apply power of a power rule:**\n$$3^{-2(a+1)} = 3^{4(8+a+1)} \cdot 3^{3(2 - a)}.$$\n\n5. **Simplify exponents:**\n- Left side: $-2(a+1) = -2a - 2$.\n- Right side: $4(8+a+1) = 4(9+a) = 36 + 4a$, and $3(2 - a) = 6 - 3a$.\n\n6. **Combine right side exponents (product of same base adds exponents):**\n$$3^{36 + 4a} \cdot 3^{6 - 3a} = 3^{(36 + 4a) + (6 - 3a)} = 3^{42 + a}.$$\n\n7. **Set exponents equal since bases are equal and nonzero:**\n$$-2a - 2 = 42 + a.$$\n\n8. **Solve for $a$:**\n$$-2a - 2 = 42 + a$$\n$$-2a - a = 42 + 2$$\n$$-3a = 44$$\n$$a = \frac{44}{-3} = -\frac{44}{3}.$$\n\n**Final answer:** $$a = -\frac{44}{3}.$$