which expression is equivalent to $\frac{(4p^{-4}q)^{-2}}{10pq^{-3}}$? assume $p\neq0,q\neq0$.\n$\frac{25p^{1…

Explanation:

Step1: Apply power - of - a - product rule

First, simplify ((4p^{-4}q)^{-2}) using the power - of - a - product rule ((ab)^n=a^n b^n) and power rule ((a^m)^n=a^{mn}). ((4p^{-4}q)^{-2}=4^{-2}p^{(-4)\times(-2)}q^{-2}=\frac{1}{16}p^{8}q^{-2})

Step2: Rewrite the original expression

The original expression (\frac{(4p^{-4}q)^{-2}}{10pq^{-3}}) becomes (\frac{\frac{1}{16}p^{8}q^{-2}}{10pq^{-3}}).

Step3: Divide by a fraction

Dividing by a fraction is the same as multiplying by its reciprocal. So (\frac{\frac{1}{16}p^{8}q^{-2}}{10pq^{-3}}=\frac{1}{16}p^{8}q^{-2}\times\frac{1}{10pq^{-3}}).

Step4: Multiply the coefficients and use exponent rules

Multiply the coefficients (\frac{1}{16}\times\frac{1}{10}=\frac{1}{160}), and for the variables, use the rule (a^m\times a^n=a^{m + n}). (p^{8}\times p^{-1}=p^{8-1}=p^{7}) and (q^{-2}\times q^{3}=q^{-2 + 3}=q). So the result is (\frac{p^{7}q}{160}).

Answer:

(\frac{p^{7}q}{160})