Question 263697
(There are errors in another solution provided.)
{{{125^(x+3)=(5^(2x))/(15625)}}}
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 {{{5^6}}}.)<br>
So we can start our solution by replacing the 125 and 15625 by 5 to the appropriate power:
{{{(5^3)^(x+3)=(5^(2x))/(5^6)}}}
Using our rules for exponents we get:
{{{5^(3x+9)=5^(2x-6)}}}
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+9 = 2x - 6}}}
Solving this we get
{{{x = -15}}}<br>
(This is the same answer as the other solution. One error in the other solution is that it has {{{125^x+125^3}}} and {{{5^(3x)+5^9}}} in several places. The plus symbols should be multiplication symbols. And later the solution assumes
{{{5^(3x)+5^9}}}
is equal to
{{{5^(3x+9)}}}.
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.)