Let's simplify the expression and then find the value of \( n \).
Given expression:
\[ \frac{a^3b^2(c^{-3})^2}{(a^2)^2} \div \frac{c}{a} \]
First, simplify the numerator and denominator separately:
Numerator:
\[ a^3b^2(c^{-3})^2 = a^3b^2c^{-6} \]
Denominator:
\[ (a^2)^2 = a^4 \]
Now, the expression becomes:
\[ \frac{a^3b^2c^{-6}}{a^4} \div \frac{c}{a} \]
To divide, subtract the exponents:
\[ a^{3-4}b^2c^{-6} \times \frac{a}{c} \]
Simplify further:
\[ a^{-1}b^2c^{-6} \times \frac{a}{c} \]
Combine the terms:
\[ \frac{b^2a}{ac^7} \]
Now, compare this with the expression \( \frac{b}{c^n} \):
\[ n = 7 \]
Therefore, the value of \( n \) is 7.
