To solve the equation [tex]\( 25^{(z-2)} = 125 \)[/tex], let's go through the steps carefully.
1. Rewrite the Base and Target in terms of Common Bases
First, observe that both 25 and 125 can be rewritten using bases of common smaller prime numbers. Notice that:
[tex]\[ 25 = 5^2 \][/tex]
[tex]\[ 125 = 5^3 \][/tex]
2. Substitute into the Equation
By substituting these expressions into the original equation, we obtain:
[tex]\[ (5^2)^{(z-2)} = 5^3 \][/tex]
3. Simplify the Exponents
Apply the power of a power property [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ 5^{2(z-2)} = 5^3 \][/tex]
4. Equate the Exponents
Since the bases are the same (both are 5), the exponents must be equal for the equation to hold true:
[tex]\[ 2(z - 2) = 3 \][/tex]
5. Solve for [tex]\( z \)[/tex]
Now we can solve the equation:
[tex]\[ 2z - 4 = 3 \][/tex]
Add 4 to both sides:
[tex]\[ 2z = 7 \][/tex]
Divide both sides by 2:
[tex]\[ z = \frac{7}{2} \][/tex]
6. Convert Fraction to Decimal
Simplify [tex]\(\frac{7}{2}\)[/tex]:
[tex]\[ z = 3.5 \][/tex]
Hence, the solution to the equation [tex]\( 25^{(z-2)} = 125 \)[/tex] is:
[tex]\[ \boxed{3.5} \][/tex]
