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my solution:

suppose a is a constructible root of the equation above.

we square both sides to get x^6 - 6x^4 + 9x^2 = 27.

since a is constructible, a^2 is constructible as well and we can turn this equation into cubic poly with rational coefficients, and it becomes y^3 - 6y^2 + 9y - 27 = 0.

If this cubic poly has a constructible root, it must have a rational root in form of m/n.

(m/n)^3 - 6(m/n)^2 + 9(m/n) = 27.

How do I proceed from here? Detailed steps would be appreciated..