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

Let's simplify the expression for the function \( g(w) \):

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

### Step 1: Factor the numerator
The numerator \( w^2 - 4w + 3 \) can be factored as:
\[
w^2 - 4w + 3 = (w - 3)(w - 1)
\]
So, the function can be rewritten as:
\[
g(w) = \frac{(w - 3)(w - 1)}{w^{\frac{3}{2}}}
\]

### Step 2: Simplify the expression
We can separate the fraction into two terms:
\[
g(w) = \frac{w - 3}{w^{\frac{3}{2}}} \times \frac{w - 1}{1}
\]

This can be written as:
\[
g(w) = \frac{w - 3}{w^{\frac{3}{2}}} \times \frac{w - 1}{1} = \frac{w - 3}{w^{\frac{3}{2}}} \times \frac{w - 1}{w^{\frac{3}{2}}}
\]

Now, simplify each term:
\[
g(w) = \frac{w}{w^{\frac{3}{2}}} - \frac{3}{w^{\frac{3}{2}}}
\]

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

So, the simplified expression is:
\[
g(w) = \frac{1}{\sqrt{w}} - \frac{3}{w^{\frac{3}{2}}}
\]

This is the final simplified form of \( g(w) \).

Would you like any details on the steps, or do you have any questions?

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Here are 5 related questions:
1. How would the simplification change if the numerator had different coefficients?
2. Can this function be further simplified or factored?
3. How would you find the derivative of \( g(w) \)?
4. What are the domain restrictions for \( g(w) \)?
5. How would the simplification change if the exponent in the denominator was different?

**Tip:** When simplifying expressions involving exponents, always try to factor and combine like terms to make the expression easier to manage.

Learn how to simplify the function g(w) = (w^2 - 4w + 3) / (w^(3/2)) step-by-step. This problem covers algebraic concepts including fractional exponents and simplification of rational functions. Suitable for high school students.

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