To determine the equivalent expression for [tex]\(\left(\frac{z^4}{6^2}\right)^{-3}\)[/tex], let's go through the process step-by-step.
1. Start with the given expression:
[tex]\[
\left(\frac{z^4}{6^2}\right)^{-3}
\][/tex]
2. Express the fraction inside the parentheses:
[tex]\[
\frac{z^4}{6^2} = \frac{z^4}{36}
\][/tex]
3. Raise the fraction to the [tex]\(-3\)[/tex]th power:
[tex]\[
\left(\frac{z^4}{36}\right)^{-3}
\][/tex]
4. Apply the property of exponents [tex]\(\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n\)[/tex]:
[tex]\[
\left(\frac{z^4}{36}\right)^{-3} = \left(\frac{36}{z^4}\right)^3
\][/tex]
5. Raise both the numerator and the denominator to the power of 3:
[tex]\[
\left(\frac{36}{z^4}\right)^3 = \frac{36^3}{(z^4)^3}
\][/tex]
6. Simplify each term:
[tex]\(\left(36\right)^3 = 36 \times 36 \times 36 = 46656\)[/tex]
[tex]\(\left(z^4\right)^3 = z^{4 \times 3} = z^{12}\)[/tex]
7. Combine the results:
[tex]\[
\left(\frac{z^4}{6^2}\right)^{-3} = \frac{46656}{z^{12}}
\][/tex]
Thus, the equivalent expression for [tex]\(\left(\frac{z^4}{6^2}\right)^{-3}\)[/tex] is [tex]\(\frac{46656}{z^{12}}\)[/tex].
Now, compare this result with the provided options:
(A) [tex]\(\frac{z}{6^{-1}}\)[/tex]
(B) [tex]\(\frac{6^6}{z^{12}}\)[/tex]
(C) [tex]\(z^{12} \cdot 6^6\)[/tex]
Clearly, the correct answer is:
(B) [tex]\(\frac{6^6}{z^{12}}\)[/tex]
Note: [tex]\(46656 = 6^6\)[/tex] because [tex]\(6^6 = 46656\)[/tex].
