1. **State the problem:** Simplify the expression $$w^{-\frac{1}{7}} \cdot w^{-\frac{12}{5}}$$ assuming all variables are positive. 2. **Recall the rule for multiplying powers with the same base:** $$a^m \cdot a^n = a^{m+n}$$ 3. **Apply the rule:** $$w^{-\frac{1}{7}} \cdot w^{-\frac{12}{5}} = w^{-\frac{1}{7} + (-\frac{12}{5})} = w^{-\frac{1}{7} - \frac{12}{5}}$$ 4. **Find a common denominator to add the exponents:** The denominators are 7 and 5, so the common denominator is 35. Convert each fraction: $$-\frac{1}{7} = -\frac{5}{35}$$ $$-\frac{12}{5} = -\frac{84}{35}$$ 5. **Add the exponents:** $$-\frac{5}{35} - \frac{84}{35} = -\frac{89}{35}$$ 6. **Rewrite the expression:** $$w^{-\frac{89}{35}}$$ 7. **Make the exponent positive by using the reciprocal:** $$w^{-\frac{89}{35}} = \frac{1}{w^{\frac{89}{35}}}$$ **Final answer:** $$\boxed{\frac{1}{w^{\frac{89}{35}}}}$$