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Solve:

[tex]\[ \left(\frac{1}{5,000}\right)^{-2z} \cdot 5,000^{-2z+2} = 5,000 \][/tex]

A. [tex]\( z = -1 \)[/tex]
B. [tex]\( z = 0 \)[/tex]
C. [tex]\( z = 1 \)[/tex]
D. No solution



Answer :

GinnyAnswer
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]

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