To factor the expression:

$4x^{3/2} - 12x^{1/2} + 8x^{-1/2}$

Step 1: Identify the greatest common factor (GCF)

We observe that all terms have a factor of $x^{-1/2}$, which is the smallest power of $x$ among the terms. Let's factor out $x^{-1/2}$ from each term:

$x^{-1/2} \left( 4x^2 - 12x + 8 \right)$

Step 2: Factor the quadratic expression inside the parentheses

Now we focus on factoring the quadratic part:

$4x^2 - 12x + 8$

First, factor out the common factor of 4:

$4 \left( x^2 - 3x + 2 \right)$

Next, we factor the quadratic $x^2 - 3x + 2$. It factors as:

$(x - 1)(x - 2)$

Step 3: Combine everything

Now substitute the factored form back into the expression:

$x^{-1/2} \cdot 4 \cdot (x - 1)(x - 2)$

So, the fully factored expression is:

$4x^{-1/2} (x - 1)(x - 2)$

Final Answer:

$4x^{-1/2} (x - 1)(x - 2)$

Would you like more details or have any questions about the process?

Related Questions:

1. How do you factor quadratics when the leading coefficient is not 1?
2. What happens when you factor out a negative term from an expression?
3. How can you check if your factoring is correct?
4. What are some techniques for factoring expressions with fractional exponents?
5. How do you handle factoring when terms have negative exponents?

Tip: When factoring terms with fractional exponents, always identify the smallest power of $x$ to factor out, which simplifies the process.