Option a:
is the correct answer.
Explanation:
The expression is ![\left(\frac{\left(3 x y^{-5}\right)^{3}}{\left(x^{-2} y^{2}\right)^{-4}}\right)^{-2}](https://tex.z-dn.net/?f=%5Cleft%28%5Cfrac%7B%5Cleft%283%20x%20y%5E%7B-5%7D%5Cright%29%5E%7B3%7D%7D%7B%5Cleft%28x%5E%7B-2%7D%20y%5E%7B2%7D%5Cright%29%5E%7B-4%7D%7D%5Cright%29%5E%7B-2%7D)
We shall simplify the expression, to determine the expression which is equivalent to ![\left(\frac{\left(3 x y^{-5}\right)^{3}}{\left(x^{-2} y^{2}\right)^{-4}}\right)^{-2}](https://tex.z-dn.net/?f=%5Cleft%28%5Cfrac%7B%5Cleft%283%20x%20y%5E%7B-5%7D%5Cright%29%5E%7B3%7D%7D%7B%5Cleft%28x%5E%7B-2%7D%20y%5E%7B2%7D%5Cright%29%5E%7B-4%7D%7D%5Cright%29%5E%7B-2%7D)
Multiplying the powers, we get,
![\left(\frac{\left3^{3} x^{3} y^{-15}\right}{x^{8} y^{-8}}\right)^{-2}](https://tex.z-dn.net/?f=%5Cleft%28%5Cfrac%7B%5Cleft3%5E%7B3%7D%20x%5E%7B3%7D%20y%5E%7B-15%7D%5Cright%7D%7Bx%5E%7B8%7D%20y%5E%7B-8%7D%7D%5Cright%29%5E%7B-2%7D)
Again multiplying the powers, we get,
![\left\frac{\left3^{-6} x^{-6} y^{30}\right}{x^{-16} y^{16}}\right](https://tex.z-dn.net/?f=%5Cleft%5Cfrac%7B%5Cleft3%5E%7B-6%7D%20x%5E%7B-6%7D%20y%5E%7B30%7D%5Cright%7D%7Bx%5E%7B-16%7D%20y%5E%7B16%7D%7D%5Cright)
Dividing the fractions, we have,
![\left\frac{\left3^{-6} y^{14}\right}{x^{-10} }\right](https://tex.z-dn.net/?f=%5Cleft%5Cfrac%7B%5Cleft3%5E%7B-6%7D%20y%5E%7B14%7D%5Cright%7D%7Bx%5E%7B-10%7D%20%7D%5Cright)
Applying the exponent rule,
, we have,
![\frac{x^{10} y^{14}}{3^{6} }](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E%7B10%7D%20y%5E%7B14%7D%7D%7B3%5E%7B6%7D%20%7D)
Hence, the expression can be written as
![\frac{x^{10} y^{14}}{729}](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E%7B10%7D%20y%5E%7B14%7D%7D%7B729%7D)
Thus, the expression which is equivalent to
is ![\frac{x^{10} y^{14}}{729}](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E%7B10%7D%20y%5E%7B14%7D%7D%7B729%7D)
Hence, Option a is the correct answer.