1. **State the problem:** Simplify the expression $$\left( \frac{3x^4}{4x^7} \right)^{-2}$$ and write the answer using only positive exponents. 2. **Recall the rules:** - When raising a fraction to a power, raise both numerator and denominator to that power: $$\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}$$. - Negative exponents mean reciprocal: $$a^{-n} = \frac{1}{a^n}$$. - When dividing powers with the same base, subtract exponents: $$\frac{x^m}{x^n} = x^{m-n}$$. 3. **Simplify inside the parentheses first:** $$\frac{3x^4}{4x^7} = \frac{3}{4} \cdot x^{4-7} = \frac{3}{4} x^{-3}$$ 4. **Rewrite the expression:** $$\left( \frac{3}{4} x^{-3} \right)^{-2}$$ 5. **Apply the power to each factor:** $$\left( \frac{3}{4} \right)^{-2} \cdot \left( x^{-3} \right)^{-2}$$ 6. **Simplify each part:** - For the fraction with negative exponent: $$\left( \frac{3}{4} \right)^{-2} = \left( \frac{4}{3} \right)^2 = \frac{4^2}{3^2} = \frac{16}{9}$$ - For the power of a power: $$\left( x^{-3} \right)^{-2} = x^{-3 \times -2} = x^6$$ 7. **Combine the results:** $$\frac{16}{9} x^6$$ **Final answer:** $$\boxed{\frac{16}{9} x^6}$$