Answer:
[tex]\large\boxed{\dfrac{1}{36a^4b^{10}}}[/tex]
Step-by-step explanation:
[tex]\left(\dfrac{\left(2a^{-3}b^4\right)^2}{\left(3a^5b\right)^{-2}}\right)^{-1}\qquad\text{use}\ a^{-1}=\dfrac{1}{a}\\\\=\dfrac{\left(3a^5b\right)^{-2}}{\left(2a^{-3}b^4\right)^2}\qquad\text{use}\ (ab)^n=a^nb^n\\\\=\dfrac{3^{-2}(a^5)^{-2}b^{-2}}{2^2(a^{-3})^2(b^4)^2}\qquad\text{use}\ (a^n)^m=a^{nm}\\\\=\dfrac{3^{-2}a^{(5)(-2)}b^{-2}}{4a^{(-3)(2)}b^{(4)(2)}}=\dfrac{3^{-2}a^{-10}b^{-2}}{4a^{-6}b^8}\qquad\text{use}\ \dfrac{a^n}{a^m}=a^{n-m}[/tex]
[tex]=\dfrac{3^{-2}}{4}a^{-10-(-6)}b^{-2-8}=\dfrac{3^{-2}}{4}a^{-4}b^{-10}\qquad\text{use}\ a^{-n}=\dfrac{1}{a^n}\\\\=\dfrac{1}{3^2}\cdot\dfrac{1}{4}\cdot\dfrac{1}{a^4}\cdot\dfrac{1}{b^{10}}=\dfrac{1}{36a^4b^{10}}[/tex]
