Let's solve the equation step-by-step:
We start with the equation given:
[tex]\[
\left(\frac{1}{5000}\right)^{-2z} \cdot 5000^{-2z + 2} = 5000
\][/tex]
First, we simplify the terms. Recall that:
[tex]\[
\left(\frac{1}{5000}\right)^{-2z} = 5000^{2z}
\][/tex]
Thus, the equation becomes:
[tex]\[
5000^{2z} \cdot 5000^{-2z + 2} = 5000
\][/tex]
We combine the exponents because the bases are the same (5000):
[tex]\[
5000^{2z + (-2z + 2)} = 5000
\][/tex]
Simplify the exponent:
[tex]\[
5000^{2z - 2z + 2} = 5000^2 = 5000
\][/tex]
Now, we have:
[tex]\[
5000^2 = 5000
\][/tex]
This simplifies to:
[tex]\[
5000 = 5000^1
\][/tex]
Clearly, [tex]\(5000^2 \neq 5000\)[/tex].
Given that there are no [tex]\(z\)[/tex] values that can satisfy the transformation process correctly, we conclude there is no solution.
So, the correct answer is:
[tex]\[
\text{no solution}
\][/tex]
