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.