1. **State the problem:** Simplify and verify the equation: $$\left(2x^{2}y^{-1} / 3z^{2}\right)^{-2} \times \left(3xz^{2} / y\right)^{3} = \frac{3^{7} z^{10}}{2^{2} x y}$$ 2. **Rewrite the expression clearly:** $$\left(\frac{2x^{2}y^{-1}}{3z^{2}}\right)^{-2} \times \left(\frac{3xz^{2}}{y}\right)^{3}$$ 3. **Apply the negative exponent rule:** $$\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$$ So, $$\left(\frac{2x^{2}y^{-1}}{3z^{2}}\right)^{-2} = \left(\frac{3z^{2}}{2x^{2}y^{-1}}\right)^{2}$$ 4. **Simplify inside the parentheses:** Note that $y^{-1} = \frac{1}{y}$, so denominator is $2x^{2} \times \frac{1}{y} = \frac{2x^{2}}{y}$. Therefore, $$\left(\frac{3z^{2}}{\frac{2x^{2}}{y}}\right)^{2} = \left(3z^{2} \times \frac{y}{2x^{2}}\right)^{2} = \left(\frac{3 y z^{2}}{2 x^{2}}\right)^{2}$$ 5. **Square the fraction:** $$\left(\frac{3 y z^{2}}{2 x^{2}}\right)^{2} = \frac{3^{2} y^{2} z^{4}}{2^{2} x^{4}} = \frac{9 y^{2} z^{4}}{4 x^{4}}$$ 6. **Simplify the second term:** $$\left(\frac{3 x z^{2}}{y}\right)^{3} = \frac{3^{3} x^{3} z^{6}}{y^{3}} = \frac{27 x^{3} z^{6}}{y^{3}}$$ 7. **Multiply the two results:** $$\frac{9 y^{2} z^{4}}{4 x^{4}} \times \frac{27 x^{3} z^{6}}{y^{3}} = \frac{9 \times 27 \times y^{2} \times z^{4} \times x^{3} \times z^{6}}{4 \times x^{4} \times y^{3}}$$ 8. **Combine like terms:** $$= \frac{243 x^{3} y^{2} z^{10}}{4 x^{4} y^{3}}$$ 9. **Cancel common factors:** $$= \frac{243 \cancel{x^{3}} y^{2} z^{10}}{4 \cancel{x^{3}} x^{1} y^{3}} = \frac{243 y^{2} z^{10}}{4 x y^{3}}$$ 10. **Simplify powers of y:** $$= \frac{243 z^{10}}{4 x y}$$ 11. **Rewrite constants:** Note that $243 = 3^{5}$ and $4 = 2^{2}$, so $$= \frac{3^{5} z^{10}}{2^{2} x y}$$ 12. **Compare with the right side:** Given right side is $$\frac{3^{7} z^{10}}{2^{2} x y}$$ Our result has $3^{5}$, but the right side has $3^{7}$. 13. **Conclusion:** The left side simplifies to $$\frac{3^{5} z^{10}}{2^{2} x y}$$ which is not equal to the right side $$\frac{3^{7} z^{10}}{2^{2} x y}$$ Therefore, the equation as given is not true unless multiplied by $3^{2}$ on the left side. **Final simplified left side:** $$\frac{3^{5} z^{10}}{2^{2} x y}$$