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solve............dunno where to start

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Okay, the first thing to do is recall properties of exponents and how to rewrite number as a single base.  What this means is if you were to rewrite 36, you could rewrite it as: 62.
Additionally if you have 81x.  Then you could rewrite as either a base of 9 or a base of 3.
recall that 81=92   so 81x=(92)x  :  from here properties of exponents tell you to multiply the exponents together, so you get:  81x=92x
If you follow the same process further you will have: 34x.
 
With this you may look at:
9x+1=27x-1   and you may see that 9 and 27 can both be rewritten as the same base.  Where:
9=32   and 27=33
 
Making the substitutions:
(32)x+1=(33)x-1
Remember that the exponents multiply.  But more importantly, since you have the same base, you may set the exponents equal to one another:
32(x+1)=33(x-1)    can only be true if the exponents are the same:
so,
2(x+1)=3(x-1)
Distributing and solving:
2x+2=3x-3
-x=-5
x=5
You may check your answer to see if this is correct and you find that it is the solution to the exponential.