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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.

Simplify the given expression: \(\frac{(a^{1/2} - 2a^{1/4} + 1) a^{1/2}}{(a^{1/8} + 1)(a^{1/8} - 1)}\), for \(0 < a < 1\).

Learn how to simplify the algebraic expression \(\frac{(a^{1/2} - 2a^{1/4} + 1) a^{1/2}}{(a^{1/8} + 1)(a^{1/8} - 1)}\), for \(0 < a < 1\). This problem involves key algebraic concepts such as fractional exponents, simplifying expressions, and using the difference of squares formula. Suitable for students in Grades 9-12.

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