The expression \(\frac{x^2 - x}{\sqrt[3]{x} - x}\) can be simplified as follows:

### Step 1: Factor the numerator
The numerator \(x^2 - x\) can be factored as:
\[
x^2 - x = x(x - 1)
\]
So the expression becomes:
\[
\frac{x(x - 1)}{\sqrt[3]{x} - x}
\]

### Step 2: Rewrite the denominator
The denominator \(\sqrt[3]{x} - x\) can be rewritten as:
\[
\sqrt[3]{x} - x = \sqrt[3]{x} - x^{1}
\]

### Step 3: Factor the difference of cubes
The expression in the denominator \(\sqrt[3]{x} - x\) is a difference of cubes, as \(x^{1} = (x^{1/3})^3\). We can use the identity for the difference of cubes:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
Here, \(a = \sqrt[3]{x}\) and \(b = 1\). Thus:
\[
\sqrt[3]{x} - x = \left(\sqrt[3]{x} - x\right)\left(\left(\sqrt[3]{x}\right)^2 + \sqrt[3]{x}\cdot x + x^2\right)
\]

However, since we already have \(\sqrt[3]{x} - x\) in the denominator, simplifying it further isn't straightforward without more specific conditions on \(x\).

### Simplification Conclusion:
Without specific conditions on \(x\), the expression cannot be simplified further directly. The expression remains:
\[
\frac{x(x - 1)}{\sqrt[3]{x} - x}
\]

**Note**: If \(x = 1\), the expression becomes undefined due to division by zero, so \(x = 1\) is a point to consider for the domain.

Would you like to explore this expression further, such as its behavior at specific points or more conditions on \(x\)?

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Here are five related questions:

1. What is the behavior of the expression \(\frac{x^2 - x}{\sqrt[3]{x} - x}\) as \(x\) approaches 1?
2. How would you find the limit of the expression as \(x\) approaches 0?
3. Can the expression be simplified using L'Hopital's Rule in calculus?
4. What is the domain of the expression \(\frac{x^2 - x}{\sqrt[3]{x} - x}\)?
5. How does the expression change if we replace \(x\) with a specific value like \(x = 8\)?

**Tip**: Factoring and recognizing special identities, like the difference of cubes, are powerful tools for simplifying algebraic expressions.

This page provides a detailed solution for simplifying the algebraic expression \frac{x^2 - x}{\sqrt[3]{x} - x}. Key concepts include factoring, radicals, and the difference of cubes identity. Explore the solution suitable for Grades 10-12.

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