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The cube root function, denoted as \( \sqrt[3]{z} \), involves finding a number that, when multiplied by itself twice, yields the original number \( z \). It simplifies the process of extracting cubes to reveal the root. In the context of the given exercise, it defines how \( x \) varies directly as \( \sqrt[3]{z} \). This means:

• As the cube root of \( z \) increases, \( x \) increases proportionally, assuming \( y \) and the constant \( k \) remain unchanged.
• Conversely, if \( \sqrt[3]{z} \) decreases, \( x \) decreases as well.

The cube root function is key in expressing relationships involving volume or three-dimensional space, as it provides a way to "reverse" the cube in mathematical terms. To compute a cube root manually might be complex, but calculators simplify the process by efficiently computing the precise value.

The constant of variation, often denoted as \( k \), is an essential component when dealing with direct and inverse variation equations. It allows us to predict the behavior of the variables based on their relational structure. In mathematical expressions, the constant remains the same as the variables vary, provided the conditions do not change.

• For direct variation, increasing one quantity results in proportional increases in the other, with \( k \) ensuring the relation's consistency.
• In inverse variation, as seen in our example \( x = k \cdot \sqrt[3]{z} / y \), the product of the variables and \( k \) stays consistent, so if one variable increases, the other must decrease to compensate.

In practical terms, \( k \) might represent a fixed rate or a constant that arises from real-world constraints. Understanding \( k \) helps clarify how strongly the variables are connected within the relationship.

Solving an equation often involves rearranging the terms so that the variable of interest is isolated on one side of the equation. Let's look at how we solve for \( y \) in the original problem:To find \( y \), start with the equation \( x = k \cdot \sqrt[3]{z} / y \).

• Multiply both sides by \( y \) to remove it from the denominator: \( y \cdot x = k \cdot \sqrt[3]{z} \).
• Next, divide both sides by \( x \) to isolate \( y \): \( y = k \cdot \sqrt[3]{z} / x \).

In such simple algebraic manipulations, keeping balance on both sides is crucial. This means whatever operation you perform on one side must be mirrored on the other. These steps allow us to express \( y \) directly in terms of the other variables, reflecting its dependency within the system. By understanding the rules that govern equation solving, students can navigate more complex mathematical problems with confidence.