1. **State the problem:** Solve the equation $$\left(\frac{1}{9}\right)^{a+1} = 81^{8+a+1} \cdot 27^{2-a}$$ for the variable $a$. 2. **Rewrite bases as powers of 3:** - $\frac{1}{9} = 9^{-1} = (3^2)^{-1} = 3^{-2}$ - $81 = 3^4$ - $27 = 3^3$ So the equation becomes: $$\left(3^{-2}\right)^{a+1} = (3^4)^{8+a+1} \cdot (3^3)^{2-a}$$ 3. **Apply power of a power rule:** $$3^{-2(a+1)} = 3^{4(8+a+1)} \cdot 3^{3(2-a)}$$ 4. **Simplify exponents:** - Left side exponent: $-2(a+1) = -2a - 2$ - Right side exponents: - $4(8+a+1) = 4(9+a) = 36 + 4a$ - $3(2 - a) = 6 - 3a$ So right side is: $$3^{36 + 4a} \cdot 3^{6 - 3a} = 3^{(36 + 4a) + (6 - 3a)} = 3^{42 + a}$$ 5. **Set exponents equal since bases are the same and nonzero:** $$-2a - 2 = 42 + a$$ 6. **Solve for $a$:** $$-2a - 2 = 42 + a$$ Add $2a$ to both sides: $$\cancel{-2a} - 2 + 2a = 42 + a + 2a \Rightarrow -2 = 42 + 3a$$ Subtract 42 from both sides: $$-2 - 42 = 42 - 42 + 3a \Rightarrow -44 = 3a$$ Divide both sides by 3: $$\frac{-44}{\cancel{3}} = \frac{3a}{\cancel{3}} \Rightarrow a = -\frac{44}{3}$$ **Final answer:** $$a = -\frac{44}{3}$$