To simplify the expression \(\frac{x^{3/2}}{x + x^{5/2} - x^2}\), we'll proceed step by step:

### Given Expression:
\[
\frac{x^{3/2}}{x + x^{5/2} - x^2}
\]

### Step 1: Factor out the greatest common factor (GCF) in the denominator.
- The terms in the denominator are \(x\), \(x^{5/2}\), and \(-x^2\).
- The smallest exponent is \(x\) or \(x^1\), so factor out \(x\) from the denominator.

\[
\frac{x^{3/2}}{x(1 + x^{3/2} - x)}
\]

### Step 2: Simplify the expression by canceling out \(x\) from the numerator and denominator.
- The \(x\) in the denominator can be canceled with \(x^{3/2}\) in the numerator.

\[
\frac{x^{3/2}}{x(1 + x^{3/2} - x)} = \frac{x^{3/2 - 1}}{1 + x^{3/2} - x} = \frac{x^{1/2}}{1 + x^{3/2} - x}
\]

### Final Simplified Expression:
\[
\frac{x^{1/2}}{1 + x^{3/2} - x}
\]

This is the simplified form of the given expression.

Would you like to go over any details, or do you have any other questions?

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**Related Questions:**
1. How can you further simplify the expression \(\frac{x^{1/2}}{1 + x^{3/2} - x}\) for specific values of \(x\)?
2. What happens to the expression when \(x = 1\)?
3. How would you expand and simplify if \(x^{3/2}\) were replaced by another term, like \(x^{5/2}\)?
4. Can you find the derivative of the simplified expression with respect to \(x\)?
5. How does this expression behave as \(x\) approaches infinity?

**Tip:** When simplifying expressions involving exponents, always look for common factors in the numerator and denominator to reduce the expression efficiently.

Learn how to simplify the expression x^(3/2) / (x + x^(5/2) - x^2) with a detailed step-by-step solution. This problem covers key algebraic concepts, including exponents and algebraic simplification. Suitable for high school students in Grades 10-12.

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