If you think about it, 1/5 is really 5^-1, which is read "five raised to the negative one."
125 becomes 5^3.
We needed to have the SAME BASE of 5 on BOTH sides of the exponential equation.
We now have this:
5^-1(x - 2) = 5^3
NEXT:
Remove the parentheses from the left side by applying the distributive rule on the exponent.
-1(x - 2) = -x + 2.
We now have this:
5^(-x + 2) = 5^3
Here comes the easy part--->bring down the exponents and solve for x.
-x + 2 = 3
-x = 3 - 2
-x = 1
x = 1/-1
x = -1
How do we know that x = -1?
Let's plug -1 for x in the ORIGINAL question given and simplify.
If we get the same answer of 125 on BOTH sides, then we will know
that x = -1.
Ready?
You were given this:
1/5^(x - 2) = 125
Let x = -1.
1/5(-1 - 2) = 125
1/5^(-3) = 125
1 divided by 1/5^3 = 125
1 divided by 1/125 = 125
125/1 = 125
125 = 125
It checks!!!
Final answer:
x = -1
Done!
(1/5)^(x-2)=125
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