Answer: [tex]\bold{c)\ \dfrac{1}{36\cdot a^{4}\cdot b^{10}}}[/tex]
Step-by-step explanation:
Use the power rule for exponents - (multiply the exponents).
Then move all of the terms that have a negative exponent to the other side of the fraction bar and change the sign of the exponent.
[tex]\bigg(\dfrac{(2a^{-3}b^4)^2}{(3a^5b)^{-2}}\bigg)^{-1}\\\\\\\\\text{distribute the exponent of -1 to both the top and bottom of the fraction:}\\=\dfrac{(2a^{-3}b^4)^{-2}}{(3a^5b)^{2}}\\\\\\\text{Now, distribute the exponent of -2 to the top and 2 to the bottom:}\\=\dfrac{2^{-2}\cdot a^{6}\cdot b^{-8}}{3^2\cdot a^{10}\cdot b^2}[/tex]
[tex]\text{Next, move }2^{-2}\ \text{and}\ b^{-8}\ \text{to the other side of the fraction}\\\text{bar and change the sign of the exponent.}\\\\=\dfrac{a^6}{2^2\cdot 3^2\cdot a^{10}\cdot b^2\cdot b^8}\\\\\\\text{Simplify }2^2\cdot3^2\ (4\cdot 9 = 36),\ \text{use the exponent rules for multiplying}\\ \text{terms that have the same base (add the exponents), and the exponent}\\\text{rules for dividing terms that have the same base (subtract the exponents).}\\\\=\dfrac{1}{36\cdot a^{10-6}\cdot b^{2+8}}[/tex]
[tex]=\dfrac{1}{36\cdot a^{4}\cdot b^{10}}[/tex]
