Let's analyze the given expression step-by-step:
The expression to simplify is:
[tex]\[
\left(\frac{3^{-6}}{7^{-3}}\right)^5
\][/tex]
First, let's simplify the expression inside the parentheses:
[tex]\[
\frac{3^{-6}}{7^{-3}}
\][/tex]
Using the property of exponents [tex]\(\frac{a^m}{b^n} = a^m \cdot b^{-n}\)[/tex], we can rewrite:
[tex]\[
\frac{3^{-6}}{7^{-3}} = 3^{-6} \cdot 7^{3}
\][/tex]
Now we raise this result to the power of 5:
[tex]\[
(3^{-6} \cdot 7^{3})^5
\][/tex]
Using the property [tex]\((a \cdot b)^m = a^m \cdot b^m\)[/tex], we split the powers:
[tex]\[
(3^{-6})^5 \cdot (7^{3})^5
\][/tex]
Now, simplify each part individually:
[tex]\[
(3^{-6})^5 = 3^{-6 \cdot 5} = 3^{-30}
\][/tex]
[tex]\[
(7^{3})^5 = 7^{3 \cdot 5} = 7^{15}
\][/tex]
Putting these simplified expressions back together, we have:
[tex]\[
3^{-30} \cdot 7^{15}
\][/tex]
This can be rewritten using a fraction to show the exponents clearly:
[tex]\[
\frac{7^{15}}{3^{30}}
\][/tex]
Therefore, the equivalent expression is:
[tex]\[
\frac{7^{15}}{3^{30}}
\][/tex]
So the correct answer is:
[tex]\[
\boxed{\frac{7^{15}}{3^{30}}}
\][/tex]
This corresponds to option (A).
