Math Problem Statement

\frac{x^2-x}{\sqrt[3]{x}-x}

Solution

The expression x2xx3x\frac{x^2 - x}{\sqrt[3]{x} - x} can be simplified as follows:

Step 1: Factor the numerator

The numerator x2xx^2 - x can be factored as: x2x=x(x1)x^2 - x = x(x - 1) So the expression becomes: x(x1)x3x\frac{x(x - 1)}{\sqrt[3]{x} - x}

Step 2: Rewrite the denominator

The denominator x3x\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 x3x\sqrt[3]{x} - x is a difference of cubes, as x1=(x1/3)3x^{1} = (x^{1/3})^3. We can use the identity for the difference of cubes: a3b3=(ab)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2) Here, a=x3a = \sqrt[3]{x} and b=1b = 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 x3x\sqrt[3]{x} - x in the denominator, simplifying it further isn't straightforward without more specific conditions on xx.

Simplification Conclusion:

Without specific conditions on xx, the expression cannot be simplified further directly. The expression remains: x(x1)x3x\frac{x(x - 1)}{\sqrt[3]{x} - x}

Note: If x=1x = 1, the expression becomes undefined due to division by zero, so x=1x = 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 xx?


Here are five related questions:

  1. What is the behavior of the expression x2xx3x\frac{x^2 - x}{\sqrt[3]{x} - x} as xx approaches 1?
  2. How would you find the limit of the expression as xx approaches 0?
  3. Can the expression be simplified using L'Hopital's Rule in calculus?
  4. What is the domain of the expression x2xx3x\frac{x^2 - x}{\sqrt[3]{x} - x}?
  5. How does the expression change if we replace xx with a specific value like x=8x = 8?

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

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Math Problem Analysis

Mathematical Concepts

Algebra
Factoring
Radicals
Difference of Cubes

Formulas

Difference of cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Theorems

Difference of cubes identity

Suitable Grade Level

Grades 10-12