1. **State the problem:** Simplify the expression $$\frac{100a^{-2}b^{2/3}}{25a^{2}b^{-1/3}}$$. 2. **Recall the rules:** - When dividing powers with the same base, subtract the exponents: $$\frac{x^m}{x^n} = x^{m-n}$$. - Negative exponents mean reciprocal: $$x^{-m} = \frac{1}{x^m}$$. - When dividing coefficients, divide normally. 3. **Divide the coefficients:** $$\frac{100}{25} = 4$$ 4. **Divide the powers of $a$:** $$a^{-2} \div a^{2} = a^{-2-2} = a^{-4}$$ 5. **Divide the powers of $b$:** $$b^{2/3} \div b^{-1/3} = b^{2/3 - (-1/3)} = b^{2/3 + 1/3} = b^{3/3} = b^{1}$$ 6. **Combine all parts:** $$4a^{-4}b^{1} = 4b a^{-4}$$ 7. **Rewrite negative exponent:** $$4b \frac{1}{a^{4}} = \frac{4b}{a^{4}}$$ **Final answer:** $$\frac{4b}{a^{4}}$$