1. **State the problem:** Simplify the expression $$\frac{ab^5 c^0}{a^0 b^{-1} c \cdot a^9 b^{-1} c}$$. 2. **Recall exponent rules:** - Any number to the zero power is 1: $$x^0 = 1$$. - When multiplying like bases, add exponents: $$x^m \cdot x^n = x^{m+n}$$. - When dividing like bases, subtract exponents: $$\frac{x^m}{x^n} = x^{m-n}$$. 3. **Simplify numerator:** Since $$c^0 = 1$$, numerator becomes $$ab^5$$. 4. **Simplify denominator:** Multiply $$a^0 b^{-1} c$$ and $$a^9 b^{-1} c$$: $$a^{0+9} b^{-1 + (-1)} c^{1+1} = a^9 b^{-2} c^2$$. 5. **Rewrite the expression:** $$\frac{ab^5}{a^9 b^{-2} c^2}$$. 6. **Apply division of exponents:** $$a^{1-9} b^{5 - (-2)} c^{0 - 2} = a^{-8} b^{7} c^{-2}$$. 7. **Rewrite negative exponents as fractions:** $$\frac{b^7}{a^8 c^2}$$. **Final answer:** $$\frac{b^7}{a^8 c^2}$$.