Select the equivalent expression.

[tex]\[
\left(\frac{x^4}{7^{-8}}\right)^{-7}=?
\][/tex]

Choose one answer:
A. [tex]\(\frac{x^{28}}{7^{56}}\)[/tex]
B. [tex]\(x^{-28} \cdot 7^{-56}\)[/tex]
C. [tex]\(\frac{x^{28}}{7^{-56}}\)[/tex]



Answer :

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]