1. **State the problem:** Simplify fully the expression $$\left( \frac{9x^4}{16y^{10}} \right)^{-\frac{1}{2}}$$. 2. **Recall the rule for negative fractional exponents:** For any nonzero number $a$, $$a^{-\frac{1}{2}} = \frac{1}{a^{\frac{1}{2}}} = \frac{1}{\sqrt{a}}$$. 3. **Apply the negative exponent rule:** $$\left( \frac{9x^4}{16y^{10}} \right)^{-\frac{1}{2}} = \frac{1}{\left( \frac{9x^4}{16y^{10}} \right)^{\frac{1}{2}}} = \frac{1}{\sqrt{\frac{9x^4}{16y^{10}}}}$$ 4. **Simplify the square root of a fraction:** $$\frac{1}{\frac{\sqrt{9x^4}}{\sqrt{16y^{10}}}} = \frac{1}{\frac{3x^2}{4y^5}}$$ 5. **Invert the denominator to simplify:** $$= \frac{1}{1} \times \frac{4y^5}{3x^2} = \frac{4y^5}{3x^2}$$ 6. **Final simplified expression:** $$\boxed{\frac{4y^5}{3x^2}}$$ 7. **Check the user's expression:** The user wrote $$\frac{9x^{-2}}{16y^{-6}}$$ which is not equivalent to the correct simplification. **Therefore, the user's simplification is incorrect.**