Certainly! Let's solve the equation [tex]\( 25^{(z-4)} = 125 \)[/tex] step by step.
1. Understanding the bases:
[tex]\[
25 = 5^2 \quad \text{and} \quad 125 = 5^3
\][/tex]
So, we can rewrite the equation using a common base of 5:
[tex]\[
(5^2)^{(z-4)} = 5^3
\][/tex]
2. Applying the power rule:
[tex]\[
5^{2(z-4)} = 5^3
\][/tex]
3. Equating the exponents:
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
2(z-4) = 3
\][/tex]
4. Solving for [tex]\( z \)[/tex]:
Let's solve the equation [tex]\( 2(z-4) = 3 \)[/tex]:
[tex]\[
2z - 8 = 3
\][/tex]
[tex]\[
2z = 11
\][/tex]
[tex]\[
z = \frac{11}{2}
\][/tex]
5. Simplifying the answer:
[tex]\[
z = 5.5
\][/tex]
Therefore, the solution to the equation [tex]\( 25^{(z-4)} = 125 \)[/tex] is [tex]\( z = 5.5 \)[/tex].
Among the given options, the correct one is:
[tex]\[
\boxed{z = 5.5}
\][/tex]
