To find the value of [tex]\( y \)[/tex] that satisfies the equation [tex]\( 125 = \left( \frac{1}{25} \right)^{y-1} \)[/tex], we'll follow these detailed steps:
1. Rewrite the equation in terms of exponents with the same base:
We know that:
[tex]\[ 125 = 5^3 \][/tex]
and
[tex]\[ \frac{1}{25} = 25^{-1} = (5^2)^{-1} = 5^{-2} \][/tex]
So, the given equation can be rewritten as:
[tex]\[ 5^3 = (5^{-2})^{y-1} \][/tex]
2. Simplify the exponent on the right-hand side:
Using the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[ (5^{-2})^{y-1} = 5^{-2(y-1)} \][/tex]
Now the equation looks like this:
[tex]\[ 5^3 = 5^{-2(y-1)} \][/tex]
3. Set the exponents equal to each other:
Since the bases are the same (both are base 5), we can set the exponents equal to each other:
[tex]\[ 3 = -2(y-1) \][/tex]
4. Solve for [tex]\( y \)[/tex]:
Start by expanding the right-hand side:
[tex]\[ 3 = -2y + 2 \][/tex]
Next, isolate the [tex]\( y \)[/tex] term by moving the [tex]\( 2 \)[/tex] to the other side:
[tex]\[ 3 - 2 = -2y \][/tex]
[tex]\[ 1 = -2y \][/tex]
Finally, solve for [tex]\( y \)[/tex] by dividing both sides by [tex]\(-2\)[/tex]:
[tex]\[ y = \frac{1}{-2} \][/tex]
[tex]\[ y = -\frac{1}{2} \][/tex]
Therefore, the value of [tex]\( y \)[/tex] that satisfies the equation [tex]\( 125 = \left( \frac{1}{25} \right)^{y-1} \)[/tex] is [tex]\(\boxed{-\frac{1}{2}}\)[/tex].
