If `9^(2x+1) = (81^(x+2))/3^x` what is x?

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`9^(2x+1)=(81^(x+2))/3^x`

Since 9=3^2 and 81=3^4, express the equation with same base.

`(3^2)^(2x+1)=((3^4)^(x+2))/3^x`

Apply the exponent property `(a^m)^n=a(^m*n)` .

`3^(2(2x+1))=3^(4(x+2))/3^x`

`3^(4x+2)=3^(4x+8)/3^x`

Then at the right side of the equation, apply the property `a^m/a^n=a^m-n` .

`3^(4x+2)=3^((4x+8)-x)`

`3^(4x+2)=3^(3x+8)`

Since both sides have the same base, to solve for x, equate the exponents equal to each other.

`4x+2=3x+8`

`4x-3x+2=3x-3x+8`

`x+2=8`

`x+2-2=8-2`

`x=6`

Hence, the solution to the given equation is `x=6` .

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