Let's simplify the given expression step-by-step.
Given:
[tex]\[ \frac{\left(4 p^{-4} q\right)^{-2}}{10 p q^{-3}} \][/tex]
First, simplify the numerator \(\left(4 p^{-4} q\right)^{-2}\):
1. Distribute the exponent \(-2\):
[tex]\[ \left(4 p^{-4} q\right)^{-2} = 4^{-2} \cdot (p^{-4})^{-2} \cdot (q)^{-2} \][/tex]
2. Simplify each term:
[tex]\[ 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \][/tex]
[tex]\[ (p^{-4})^{-2} = p^{8} \][/tex]
[tex]\[ q^{-2} = q^{-2} \][/tex]
Thus, we have:
[tex]\[ \left(4 p^{-4} q\right)^{-2} = \frac{1}{16} \cdot p^8 \cdot q^{-2} = \frac{p^8}{16 q^2} \][/tex]
Next, revisit the entire expression with the updated numerator:
[tex]\[ \frac{\frac{p^8}{16 q^2}}{10 p q^{-3}} \][/tex]
Simplify the denominator:
[tex]\[ 10 p q^{-3} = 10 p \cdot \frac{1}{q^3}= \frac{10 p}{q^3} \][/tex]
Therefore, the expression becomes:
[tex]\[ \frac{\frac{p^8}{16 q^2}}{\frac{10 p}{q^3}} \][/tex]
Use the property of division (divide by a fraction by multiplying by its reciprocal):
[tex]\[ \frac{p^8}{16 q^2} \cdot \frac{q^3}{10 p} = \frac{p^8 q^3}{16 q^2 \cdot 10 p} \][/tex]
Simplify by cancelling common factors:
[tex]\[ \frac{p^8 q^3}{160 p q^2} = \frac{p^{8-1} q^{3-2}}{160} = \frac{p^7 q}{160} \][/tex]
Thus, the equivalent expression to the given one is:
[tex]\[ \boxed{\frac{p^7 q}{160}} \][/tex]
So the correct answer among the provided options is:
[tex]\[ \frac{p^7 q}{160} \][/tex]
