g\left(w\right)=\frac{w^2-4w+3}{w^{\frac{3}{2}}}

The function given is:

\[
g(w) = \frac{w^2 - 4w + 3}{w^{\frac{3}{2}}}
\]

### Simplifying the Expression

To simplify the function, we can split the terms in the numerator by the denominator:

\[
g(w) = \frac{w^2}{w^{\frac{3}{2}}} - \frac{4w}{w^{\frac{3}{2}}} + \frac{3}{w^{\frac{3}{2}}}
\]

Now, simplify each term:

1. \(\frac{w^2}{w^{\frac{3}{2}}} = w^{2 - \frac{3}{2}} = w^{\frac{4}{2} - \frac{3}{2}} = w^{\frac{1}{2}}\)
2. \(\frac{4w}{w^{\frac{3}{2}}} = 4w^{1 - \frac{3}{2}} = 4w^{\frac{2}{2} - \frac{3}{2}} = 4w^{-\frac{1}{2}}\)
3. \(\frac{3}{w^{\frac{3}{2}}} = 3w^{-\frac{3}{2}}\)

So, the simplified form of the function is:

\[
g(w) = w^{\frac{1}{2}} - 4w^{-\frac{1}{2}} + 3w^{-\frac{3}{2}}
\]

### Next Steps or Questions?

Would you like to analyze the behavior of this function further, such as finding its domain, derivatives, or critical points?

Here are some related questions:

1. How do you find the domain of the simplified function \(g(w)\)?
2. What is the first derivative of \(g(w)\), and how do you find the critical points?
3. How does \(g(w)\) behave as \(w\) approaches 0 or infinity?
4. How do you find the concavity of \(g(w)\) using the second derivative?
5. What is the significance of the exponents in \(g(w)\) with respect to its graph?

**Tip:** When simplifying functions involving exponents, always remember the laws of exponents. This can help make complex expressions more manageable.

Learn how to simplify the rational function g(w) = (w^2 - 4w + 3) / (w^(3/2)) step-by-step. This problem covers functions, exponents, and rational function simplification. Suitable for students in Grades 10-12.

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