Math Problem Statement
\frac{x^2-x}{\sqrt[3]{x}-x}
Solution
The expression can be simplified as follows:
Step 1: Factor the numerator
The numerator can be factored as: So the expression becomes:
Step 2: Rewrite the denominator
The denominator 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 is a difference of cubes, as . We can use the identity for the difference of cubes: Here, and . 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 in the denominator, simplifying it further isn't straightforward without more specific conditions on .
Simplification Conclusion:
Without specific conditions on , the expression cannot be simplified further directly. The expression remains:
Note: If , the expression becomes undefined due to division by zero, so 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 ?
Here are five related questions:
- What is the behavior of the expression as approaches 1?
- How would you find the limit of the expression as approaches 0?
- Can the expression be simplified using L'Hopital's Rule in calculus?
- What is the domain of the expression ?
- How does the expression change if we replace with a specific value like ?
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
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