What is the solution to the equation 81^(x + 2) = 9^(2x)?

To solve the equation ( 81^{(x + 2)} = 9^{(2x)} ), we can express both sides of the equation in terms of powers of 3, because both 81 and 9 are powers of 3. We know that ( 81 = 3^4 ) and ( 9 = 3^2 ).

Rewriting the equation gives us:

[

(3^4)^{(x + 2)} = (3^2)^{(2x)}

]

This simplifies to:

[

3^{4(x + 2)} = 3^{2(2x)}

]

Since the bases are the same, we can set the exponents equal to each other:

[

4(x + 2) = 2(2x)

]

Expanding both sides leads to:

[

4x + 8 = 4x

]

Now, if we subtract ( 4x ) from both sides, we get:

[

8 = 0

]

This statement is a contradiction, meaning there are no values of ( x ) that can satisfy the original equation. Therefore, this indicates the scenario in