81, 27 and 9 are all powers of 3.
Let's write them in terms of 3.
81 = 3^4
Therefore we can write, [tex]81^{x-2}=3^{4(x-2)}[/tex]
and since 27 = 3^3 we can write, [tex]27^x=3^{3x}[/tex]
and 9 is 3^2.
So our equation looks like this,[tex]\dfrac{3^{4(x-2)}}{3^{3x}}=3^2[/tex]
Applying our exponent rule, division to subtraction, gives us,[tex]3^{4(x-2)-3x}=3^2[/tex]
And from this point, since the BASES are the same,
we can simply equate the exponents:
4(x-2)-3x = 2
and then solve for x.
