which expression is equivalent to $\frac{(3m^{-2}n)^{-3}}{6mn^{-2}}$? assume $m\neq0,n\neq0$.\n$\frac{m^{5}}{…

which expression is equivalent to $\frac{(3m^{-2}n)^{-3}}{6mn^{-2}}$? assume $m\neq0,n\neq0$.\n$\frac{m^{5}}{162n}$\n$\frac{1}{2m^{3}n}$\n$\frac{8m^{9}}{n^{9}}$\n$\frac{4m^{8}}{3n^{3}}$

which expression is equivalent to $\frac{(3m^{-2}n)^{-3}}{6mn^{-2}}$? assume $m\neq0,n\neq0$.\n$\frac{m^{5}}{162n}$\n$\frac{1}{2m^{3}n}$\n$\frac{8m^{9}}{n^{9}}$\n$\frac{4m^{8}}{3n^{3}}$

Answer

Explanation:

Step1: Simplify the numerator

Apply the power - of - a - product rule ((ab)^n=a^n b^n) and power - of - a - power rule ((a^m)^n=a^{mn}) to ((3m^{-2}n)^{-3}). ((3m^{-2}n)^{-3}=3^{-3}m^{(-2)\times(-3)}n^{-3}=\frac{m^{6}}{27n^{3}})

Step2: Divide by the denominator

We have (\frac{\frac{m^{6}}{27n^{3}}}{6mn^{-2}}), which is equivalent to (\frac{m^{6}}{27n^{3}}\times\frac{1}{6mn^{-2}}) (dividing by a fraction is multiplying by its reciprocal). Using the rule (\frac{a^m}{a^n}=a^{m - n}) and (a^m\times a^n=a^{m + n}), we get (\frac{m^{6-1}}{27\times6n^{3-2}}=\frac{m^{5}}{162n})

Answer:

(\frac{m^{5}}{162n})