To evaluate the expression [tex]\(\frac{(z^2)^4}{5z^5}\)[/tex] when [tex]\(z = 5\)[/tex], follow these steps:
1. Substitute [tex]\(z = 5\)[/tex] into the expression:
[tex]\[
\frac{(5^2)^4}{5 \cdot 5^5}
\][/tex]
2. Simplify the numerator [tex]\((5^2)^4\)[/tex]:
[tex]\[
5^2 = 25
\][/tex]
[tex]\[
(25)^4 = 25^4
\][/tex]
Rewriting [tex]\(25\)[/tex] as [tex]\(5^2\)[/tex], we get:
[tex]\[
25^4 = (5^2)^4 = 5^{8}
\][/tex]
Thus, the numerator simplifies to [tex]\(5^8\)[/tex].
3. Simplify the denominator [tex]\(5 \cdot 5^5\)[/tex]:
[tex]\[
5 \cdot 5^5
\][/tex]
Using the properties of exponents, this can be simplified to:
[tex]\[
5^1 \cdot 5^5 = 5^{1+5} = 5^6
\][/tex]
4. Simplify the fraction [tex]\(\frac{5^8}{5^6}\)[/tex]:
Using the property of exponents [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex] where [tex]\(a \neq 0\)[/tex]:
[tex]\[
\frac{5^8}{5^6} = 5^{8-6} = 5^2
\][/tex]
5. Calculate [tex]\(5^2\)[/tex]:
[tex]\[
5^2 = 25
\][/tex]
Therefore, the simplified value of the expression [tex]\(\frac{(z^2)^4}{5z^5}\)[/tex] when [tex]\(z = 5\)[/tex] is [tex]\(\boxed{25}\)[/tex].
