Option a: [tex]\frac{m^{5} }{162n}[/tex] is the equivalent expression.
Explanation:
The expression is [tex]\frac{(3m^{-2} n)^{-3}}{6mn^{-2} }[/tex] where [tex]m\neq 0, n\neq 0[/tex]
Let us simplify the expression, to determine which expression is equivalent from the four options.
Multiplying the powers, we get,
[tex]\frac{3^{-3}m^{6} n^{-3}}{6mn^{-2} }[/tex]
Cancelling the like terms, we have,
[tex]\frac{3^{-3}m^{5} n^{-1}}{6 }[/tex]
This equation can also be written as,
[tex]\frac{m^{5}}{3^{3}6 n^{1} }[/tex]
Multiplying the terms in denominator, we have,
[tex]\frac{m^{5} }{162n}[/tex]
Thus, the expression which is equivalent to [tex]\frac{(3m^{-2} n)^{-3}}{6mn^{-2} }[/tex] is [tex]\frac{m^{5} }{162n}[/tex]
Hence, Option a is the correct answer.
