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When solving radical equations involving cube roots, a key step is to eliminate the radical by raising both sides of the equation to the power of three, which is known as 'cube root elimination'. This process helps in transforming a radical equation into a more familiar polynomial form.

In the given example, the cube root \( (3x - 6)^{1/3} \) is eliminated by cubing both sides of the equation. Effectively, \( [(3x - 6)^{1/3}]^3 = 3^3 \) simplifies to \( 3x - 6 = 27 \) because \( (a^{1/3})^3 = a \) for any value of 'a'. This step is crucial because it converts the equation into a linear form that is much easier to solve.

Isolating the radical term on one side of the equation is an essential step before you can eliminate the radical. This process involves moving all other terms to the opposite side of the equation. By doing so, the equation is set up properly for the elimination of the cube root.

In the provided exercise, to isolate the term \( (3x - 6)^{1/3} \) the equation is rewritten and the constant term 5 is moved to the other side, giving us \( (3x - 6)^{1/3} = 3 \) by subtracting 5 from both sides. By keeping the radical term by itself, the equation is prepped for the cube root elimination that follows.

After obtaining a potential solution for an algebraic equation, it is paramount to verify that it indeed satisfies the original equation. Checking the solution involves substituting the found value back into the original equation and calculating to confirm that both sides equate.

Using our example, after deducing that \( x = 11 \), this value is substituted back into the original equation \( (3x - 6)^{1/3} + 5 = 8 \). If after the substitution the simplified equation yields a true statement, in this case, \( 8 = 8 \), then the solution can be verified as correct. The checking process ensures that no extraneous solutions are included, which are results that emerge from the steps of solving the equation but do not actually solve the original equation.