91Ó°ÊÓ

The isolation method is a fundamental technique used in solving equations. The main aim is to separate the variable from other elements in the equation. This makes dealing with complex algebraic expressions much more manageable.
In our problem, we started with the equation \( \sqrt[3]{2x-4} = 6 \). Notice that the cube root, \( \sqrt[3]{2x-4} \), is already isolated—meaning it stands alone on one side of the equation without other terms.
This situation simplifies our work because we can directly apply further operations to both sides of the equation without dealing with additional steps of isolation. Thus, recognizing when a term is already isolated is just as important as the steps to isolate it, making it a crucial skill in solving equations effectively.

To eliminate a cube root in an equation, one technique is cubing both sides. This action effectively removes the cube root, leaving a simpler expression to work with.
In our equation \( \sqrt[3]{2x - 4} = 6 \), we respond by cubing both sides:

• The left side: \( (\sqrt[3]{2x - 4})^3 = 2x - 4 \)
• The right side: \( 6^3 = 216 \)

Thus, we transform the equation into \( 2x - 4 = 216 \).
Cubing both sides is effective because it reverts the cube root operation, simplifying the equation and paving the way for further solving processes. This strategy is applicable in many algebraic contexts where roots are present.

Once the cube root is eradicated and the equation is simplified, the next step is solving for \( x \). This involves basic algebraic operations to isolate the variable.
Starting from \( 2x - 4 = 216 \), the steps are as follows:

• Add 4 to both sides to counter the \(-4\) on the left: \( 2x - 4 + 4 = 216 + 4 \).
• Simplify to get \( 2x = 220 \).
• Divide both sides by 2 to solve for \( x \): \( \frac{2x}{2} = \frac{220}{2} \).

This leads to \( x = 110 \).
The process of solving for \( x \) relies on performing operations that undo the transformations associated with \( x \). Each step cleverly strips down the equation until \( x \) is entirely isolated, delivering a clear solution.