The given expression is:

$\frac{ \left( a^{\frac{1}{2}} - 2a^{\frac{1}{4}} + 1 \right) \cdot a^{\frac{1}{2}} }{ \left( a^{\frac{1}{8}} + 1 \right) \left( a^{\frac{1}{8}} - 1 \right) }, \quad \text{for } 0 < a < 1$

We need to simplify this expression. Let's go through it step by step:

Step 1: Analyze the numerator

The numerator is: $\left( a^{\frac{1}{2}} - 2a^{\frac{1}{4}} + 1 \right) \cdot a^{\frac{1}{2}}$ First, distribute $a^{\frac{1}{2}}$ over the terms in parentheses: $a^{\frac{1}{2}} \cdot a^{\frac{1}{2}} - 2a^{\frac{1}{4}} \cdot a^{\frac{1}{2}} + a^{\frac{1}{2}} \cdot 1$ This simplifies further as: $a^1 - 2a^{\frac{3}{4}} + a^{\frac{1}{2}}$ Thus, the numerator becomes: $a - 2a^{\frac{3}{4}} + a^{\frac{1}{2}}$

Step 2: Analyze the denominator

The denominator is: $\left( a^{\frac{1}{8}} + 1 \right) \left( a^{\frac{1}{8}} - 1 \right)$ This is a difference of squares: $\left( a^{\frac{1}{8}} \right)^2 - 1^2 = a^{\frac{1}{4}} - 1$ Thus, the denominator simplifies to: $a^{\frac{1}{4}} - 1$

Step 3: Final simplified expression

Now, the entire expression becomes: $\frac{a - 2a^{\frac{3}{4}} + a^{\frac{1}{2}}}{a^{\frac{1}{4}} - 1}$

This is the simplified form of the given expression.

Would you like further details or explanations on any step?

Related Questions:

1. How do you handle exponents in more complex algebraic expressions?
2. Can the numerator of the simplified expression be factored further?
3. How would this expression change if $a > 1$?
4. What happens if $a = 0$ in this context?
5. How can the denominator be factored or simplified in another scenario?

Tip:

When simplifying algebraic expressions with fractional exponents, always break down each step to ensure clarity, especially when distributing terms.