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Taking the cube root is a method used to determine a number that, when multiplied by itself three times, results in the original number. In the context of algebraic equations, extracting the cube root can simplify the solution when working with expressions involving cube terms.
To take the cube root of a number or expression:

• Identify the quantity for which you intend to find the cube root.
• The cube root is represented by the symbol \(\sqrt[3]{\cdot}\).

In the provided equation, once the equation is rearranged, we apply the cube root to both sides. This step is crucial in isolating the variable \(y\) in terms of \(x\). Calculating \(y = \sqrt[3]{3xy - x^{3}}\) provides a relationship between the variables that might be useful for solving further problems or analyzing the equation over different ranges of values.

Equation simplification is a fundamental skill in algebra that involves reducing equations to their simplest form. This process makes it easier to work with and solve equations. Simplifying equations can include a variety of techniques:

• Combining like terms.
• Rearranging expressions.
• Factoring numbers or expressions.
• Canceling terms that appear on both sides of the equation.

In our exercise, simplification was shown in the step where the equation was rearranged to isolate \(y^3\) on one side of the equation, transforming it into \(y^{3} = 3xy - x^{3}\). This rearrangement is a key part of simplification, as it sets up the equation for an easier solution when you proceed to the next step of taking the cube root.

Isolating variables is a crucial technique in solving algebraic equations. It involves rearranging the equation so that one variable is isolated on one side of the equation. This technique has several benefits:

• Clearly defines one variable in terms of others, aiding problem analysis.
• Facilitates substitution or solution finding for specific values.

In the task, isolating \(y^3\) was our main goal. We moved terms across the equation to bring \(y^3\) alone on the left side, resulting in \(y^{3} = 3xy - x^{3}\). Once isolated, the equation can be manipulated further to solve for \(y\) by applying the cube root. This approach highlights how isolating variables can simplify complex solutions and make problems more manageable.