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

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Question 263697: 125^(x+3)=(5^(2x))/(15625)
Found 2 solutions by mananth, jsmallt9:
Answer by mananth(16946) About Me  (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) About Me  (Show Source):
You can put this solution on YOUR website!
(There are errors in another solution provided.)
125%5E%28x%2B3%29=%285%5E%282x%29%29%2F%2815625%29
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 5%5E6.)

So we can start our solution by replacing the 125 and 15625 by 5 to the appropriate power:
%285%5E3%29%5E%28x%2B3%29=%285%5E%282x%29%29%2F%285%5E6%29
Using our rules for exponents we get:
5%5E%283x%2B9%29=5%5E%282x-6%29
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:
3x%2B9+=+2x+-+6
Solving this we get
x+=+-15

(This is the same answer as the other solution. One error in the other solution is that it has 125%5Ex%2B125%5E3 and 5%5E%283x%29%2B5%5E9 in several places. The plus symbols should be multiplication symbols. And later the solution assumes
5%5E%283x%29%2B5%5E9
is equal to
5%5E%283x%2B9%29.
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.)