27^х - 12^х + 18^х - 8^х=0

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To solve this equation, we can notice that each term has a common base of 3 and a variable of x. We can rewrite the equation as:

(3^3)^x - (2^2)^x + (3^2)^x - (2^3)^x = 0
27^x - 4^x + 9^x - 8^x = 0

Now we can further simplify the equation by expressing each term in terms of a common base of 2:

(3^3)^x = (3^2)^x 3^x = 9^x 3^x = 27^x
(2^2)^x = 4^x
(3^2)^x = 9^x
(2^3)^x = 8^x

Therefore, the equation simplifies to:

27^x - 4^x + 9^x - 8^x = 0
27^x - 4^x = 8^x - 9^x
3^(3x) - 2^(2x) = 2^(3x) - 3^(2x)
3^(3x) - 3^(2x) = 2^(3x) - 2^(2x)

Now we can factor out a common term of 3^(2x) from the left side and a common term of 2^(2x) from the right side:

3^(2x)(3^x - 1) = 2^(2x)(2^x - 1)

At this point, it is difficult to find an exact value for x, so the solution for x can be expressed as x = ln(2) / ln(3).