1. **State the problem:** Simplify the expression $$\left[\frac{(2.1a^2bc)^2 \left(-\frac{35}{4} a^4 b^{-1} c^{-2}\right)^{-1}}{\left(-2\frac{1}{2} a^5 b^0 c^{-7}\right)^{-2} \left(-3\frac{3}{4} ab^3 c^{-2}\right)^2}\right]^{-\frac{3}{4}}$$ and leave no negative exponents. 2. **Rewrite mixed numbers as improper fractions:** $$-2\frac{1}{2} = -\frac{5}{2}, \quad -3\frac{3}{4} = -\frac{15}{4}$$ 3. **Simplify each part inside the brackets:** - Numerator part 1: $$(2.1a^2bc)^2 = 2.1^2 a^{2\times 2} b^{1\times 2} c^{1\times 2} = 4.41 a^4 b^2 c^2$$ - Numerator part 2: $$\left(-\frac{35}{4} a^4 b^{-1} c^{-2}\right)^{-1} = -\frac{4}{35} a^{-4} b^{1} c^{2}$$ - Denominator part 1: $$\left(-\frac{5}{2} a^5 b^0 c^{-7}\right)^{-2} = \left(-\frac{5}{2}\right)^{-2} a^{-10} b^{0} c^{14} = \frac{4}{25} a^{-10} c^{14}$$ - Denominator part 2: $$\left(-\frac{15}{4} ab^3 c^{-2}\right)^2 = \left(-\frac{15}{4}\right)^2 a^{2} b^{6} c^{-4} = \frac{225}{16} a^{2} b^{6} c^{-4}$$ 4. **Combine numerator:** $$4.41 a^4 b^2 c^2 \times -\frac{4}{35} a^{-4} b^{1} c^{2} = 4.41 \times -\frac{4}{35} a^{4-4} b^{2+1} c^{2+2} = -\frac{17.64}{35} b^{3} c^{4}$$ 5. **Combine denominator:** $$\frac{4}{25} a^{-10} c^{14} \times \frac{225}{16} a^{2} b^{6} c^{-4} = \frac{4}{25} \times \frac{225}{16} a^{-10+2} b^{6} c^{14-4} = \frac{900}{400} a^{-8} b^{6} c^{10} = \frac{9}{4} a^{-8} b^{6} c^{10}$$ 6. **Form the fraction inside the brackets:** $$\frac{-\frac{17.64}{35} b^{3} c^{4}}{\frac{9}{4} a^{-8} b^{6} c^{10}} = -\frac{17.64}{35} \times \frac{4}{9} a^{8} b^{3-6} c^{4-10} = -\frac{17.64 \times 4}{35 \times 9} a^{8} b^{-3} c^{-6}$$ 7. **Simplify the coefficient:** $$\frac{17.64 \times 4}{35 \times 9} = \frac{70.56}{315} = \frac{70.56 \div 7.056}{315 \div 7.056} = \frac{10}{44.64} \approx 0.2237$$ But better to keep fraction exact: $$17.64 = \frac{441}{25} \Rightarrow \frac{441}{25} \times 4 = \frac{1764}{25}$$ Denominator: $$35 \times 9 = 315$$ So coefficient: $$\frac{1764}{25} \div 315 = \frac{1764}{25} \times \frac{1}{315} = \frac{1764}{7875}$$ Simplify numerator and denominator by 3: $$\frac{1764 \div 3}{7875 \div 3} = \frac{588}{2625}$$ Divide numerator and denominator by 21: $$\frac{588 \div 21}{2625 \div 21} = \frac{28}{125}$$ So coefficient is $-\frac{28}{125}$. 8. **Rewrite the fraction inside the brackets:** $$-\frac{28}{125} a^{8} b^{-3} c^{-6}$$ 9. **Apply the outer exponent $-\frac{3}{4}$:** $$\left(-\frac{28}{125} a^{8} b^{-3} c^{-6}\right)^{-\frac{3}{4}} = \left(-\frac{28}{125}\right)^{-\frac{3}{4}} a^{8 \times -\frac{3}{4}} b^{-3 \times -\frac{3}{4}} c^{-6 \times -\frac{3}{4}}$$ $$= \left(-\frac{28}{125}\right)^{-\frac{3}{4}} a^{-6} b^{\frac{9}{4}} c^{\frac{9}{2}}$$ 10. **Simplify the coefficient:** $$\left(-\frac{28}{125}\right)^{-\frac{3}{4}} = \left(-1\right)^{-\frac{3}{4}} \times \left(\frac{28}{125}\right)^{-\frac{3}{4}} = -1 \times \left(\frac{125}{28}\right)^{\frac{3}{4}}$$ 11. **Final simplified expression:** $$- \left(\frac{125}{28}\right)^{\frac{3}{4}} a^{-6} b^{\frac{9}{4}} c^{\frac{9}{2}} = - \left(\frac{125}{28}\right)^{\frac{3}{4}} \frac{b^{\frac{9}{4}} c^{\frac{9}{2}}}{a^{6}}$$ 12. **Rewrite to remove negative exponents:** $$= - \left(\frac{125}{28}\right)^{\frac{3}{4}} \frac{b^{\frac{9}{4}} c^{\frac{9}{2}}}{a^{6}}$$ This is the fully simplified expression with no negative exponents.