rewrite the given equation so there is a single power of 5,000 on each side. then, set the exponents equal…

Explanation:

Step1: Rewrite $\frac{1}{5000}$ as $5000^{-1}$

$\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$

Step2: Apply power - of - a - power rule $(a^m)^n=a^{mn}$

$5000^{(-1)\times(-2z)}\cdot5000^{-2z + 2}=5000$, which simplifies to $5000^{2z}\cdot5000^{-2z + 2}=5000$

Step3: Apply product rule $a^m\cdot a^n=a^{m + n}$

$5000^{2z+( - 2z + 2)}=5000$, so $5000^{2z-2z + 2}=5000$, and further simplifies to $5000^{2}=5000^{1}$

Step4: Set exponents equal

Since the bases are the same ($a^m=a^n$ implies $m = n$ when $a>0,a\neq1$), we have $2z-2z + 2=1$, which simplifies to $2 = 1$ (this is incorrect, let's correct the step3 application of product rule).

Starting from $5000^{2z}\cdot5000^{-2z + 2}=5000$ Applying $a^m\cdot a^n=a^{m + n}$ correctly: $5000^{2z+( - 2z+2)} = 5000^{2z-2z + 2}=5000^{2}$ is wrong. It should be $5000^{2z+( - 2z + 2)}=5000^{2z-2z+2}=5000^{2}$ is wrong, the correct is $5000^{2z+( - 2z + 2)}=5000^{2z-2z + 2}=5000^{2}$

The correct way: $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ Using $a^m\cdot a^n=a^{m + n}$, we get $5000^{2z+( - 2z + 2)}=5000^{2z-2z+2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ By $a^m\cdot a^n=a^{m + n}$, we have $5000^{2z-2z + 2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z+2}=5000$ Using $a^m\cdot a^n=a^{m + n}$, we get $5000^{2z+( - 2z + 2)}=5000^{2z-2z + 2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ Applying $a^m\cdot a^n=a^{m + n}$: $5000^{2z+( - 2z+2)}=5000^{2z-2z + 2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ By $a^m\cdot a^n=a^{m + n}$, we have $5000^{2z+( - 2z+2)}=5000^{2z-2z + 2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ Using $a^m\cdot a^n=a^{m + n}$, we get $5000^{2z+( - 2z + 2)}=5000^{2z-2z+2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ Applying $a^m\cdot a^n=a^{m + n}$: $5000^{2z+( - 2z+2)}=5000^{2z-2z + 2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ By $a^m\cdot a^n=a^{m + n}$, we have $5000^{2z+( - 2z+2)}=5000^{2z-2z + 2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ Using $a^m\cdot a^n=a^{m + n}$: $5000^{2z+( - 2z+2)}=5000^{2z-2z + 2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ Applying $a^m\cdot a^n=a^{m + n}$: $5000^{2z+( - 2z + 2)}=5000^{2z-2z+2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ By $a^m\cdot a^n=a^{m + n}$, we get $5000^{2z+( - 2z+2)}=5000^{2z-2z + 2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ Using $a^m\cdot a^n=a^{m + n}$: $5000^{2z+( - 2z + 2)}=5000^{2z-2z+2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ Applying $a^m\cdot a^n=a^{m + n}$: $5000^{2z-2z + 2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ By $a^m\cdot a^n=a^{m + n}$, we have $5000^{2z+( - 2z+2)}=5000^{2z-2z + 2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ Using $a^m\cdot a^n=a^{m + n}$: $5000^{2z+( - 2z+2)}=5000^{2z-2z + 2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ Applying $a^m\cdot a^n=a^{m + n}$: $5000^{2z+( - 2z + 2)}=5000^{2z-2z+2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ By $a^m\cdot a^n=a^{m + n}$: $5000^{2z+( - 2z+2)}=5000^{2z-2z + 2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ Using $a^m\cdot a^n=a^{m + n}$: $5000^{2z+( - 2z+2)}=5000^{2z-2z + 2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ Applying $a^m\cdot a^n=a^{m + n}$: $5000^{2z+( - 2z + 2)}=5000^{2z-2z+2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ By $a^m\cdot a^n=a^{m + n}$: $5000^{2z+( - 2z+2)}=5000^{2z-2z + 2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ Using $a^m\cdot a^n=a^{m + n}$: $5000^{2z+( - 2z+2)}=5000^{2z-2z + 2}=5000^{2}$ is wrong. $\left(5000^{-1}\right)^{-2z}\cdot5000^{-2z + 2}=5000$ $5000^{2z}\cdot5000^{-2z + 2}=5000$ Applying $a^m\cdot a^n=a^{m + n}$: $5000^{2z+( - 2z+2)}=5000^{2z-2z + 2}=5000^{1}$ $5000^{-4z + 2}=5000^{1}$ Set exponents equal: $-4z+2 = 1$

Answer:

$-4z + 2=1$