Let's simplify the given expression step-by-step.
The given expression is:
[tex]\[
\left(\frac{x^4}{7^{-8}}\right)^{-7}
\][/tex]
Firstly, simplify the fraction inside the parentheses:
[tex]\[
\frac{x^4}{7^{-8}}
\][/tex]
Since [tex]\(7^{-8}\)[/tex] in the denominator is the same as [tex]\(7^8\)[/tex] in the numerator, we rewrite the expression as:
[tex]\[
\frac{x^4}{1} \cdot 7^8 = x^4 \cdot 7^8
\][/tex]
So, the expression now becomes:
[tex]\[
(x^4 \cdot 7^8)^{-7}
\][/tex]
Now, apply the exponent [tex]\(-7\)[/tex] to each part inside the parentheses:
[tex]\[
(x^4)^{-7} \cdot (7^8)^{-7}
\][/tex]
To simplify this further, use the power rule [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
x^{4 \cdot -7} \cdot 7^{8 \cdot -7}
\][/tex]
This gives us:
[tex]\[
x^{-28} \cdot 7^{-56}
\][/tex]
The resulting expression is:
[tex]\[
x^{-28} \cdot 7^{-56}
\][/tex]
Therefore, the equivalent expression is:
[tex]\[
(B) \quad x^{-28} \cdot 7^{-56}
\][/tex]
So, the correct answer is:
[tex]\[
(B) \quad x^{-28} \cdot 7^{-56}
\][/tex]
