The function you've provided is:

\[
f(x) = x \sqrt{x} = x \cdot x^{1/2} = x^{3/2}
\]

This can be simplified as:

\[
f(x) = x^{3/2}
\]

Would you like to analyze or explore any specific aspects of this function (like derivatives, integrals, etc.)? Here are some possible directions we can explore:

1. What is the derivative of \( f(x) = x^{3/2} \)?
2. What is the integral of \( f(x) = x^{3/2} \)?
3. What is the behavior of \( f(x) \) as \( x \to \infty \)?
4. What is the behavior of \( f(x) \) as \( x \to 0 \)?
5. Can we find any critical points or extrema for this function?

**Tip**: When dealing with fractional exponents, remember that \( x^a \) can be interpreted as the function \( x \) raised to a power \( a \), and this helps in both differentiation and integration.

Explore the function f(x) = x√x, simplified as f(x) = x^(3/2). This solution covers algebraic manipulation of fractional exponents and introduces key calculus concepts such as derivatives, integrals, and behavior at infinity. Suitable for Grades 9-12.

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