SOLUTION: 125^(x+3)=(5^(2x))/(15625)

Question 263697: 125^(x+3)=(5^(2x))/(15625)
Found 2 solutions by mananth, jsmallt9:
Answer by mananth(16946) (Show Source): You can put this solution on YOUR website!
125^(x+3)=(5^(2x))/(15625)
a^m*a^n=a^(m+n)
125^x+125^3= 5^2x/ 15625
(5^3)^x +(5^3)^3 = 5^2x/15625
5^3x +5^9=5^2x/(25)^3
5^3x +5^9 = 5^2x / 5^6
5^3x +5^9 = 5^(2x-6)
3x+9=2x-6
x=-15
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
(There are errors in another solution provided.)

As in that other solution, the key to a relative simple solution is to recognize that 125 and 15625 are powers of 5. (I don't immediately recognize that 15625 is a power of 5 but, since it ends in 5, it could be. And we can find out which power of 5 it is, if any, by finding successive powers of 5. It turns out to be .)

So we can start our solution by replacing the 125 and 15625 by 5 to the appropriate power:

Using our rules for exponents we get:

We have a power of 5 equal to another power of 5. The only way this can be true is if the exponents are equal:

Solving this we get

(This is the same answer as the other solution. One error in the other solution is that it has and in several places. The plus symbols should be multiplication symbols. And later the solution assumes

is equal to
.
But they are not equal. These two errors happen to cancel each other out which is why the answer accidentally works out to be correct.)
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