Answer :

Sure, let's rationalize the denominator for the fraction [tex]\(\frac{\sqrt{2} - 1}{3 \sqrt{\sqrt{2}} - 2}\)[/tex].

### Step 1: Simplify the Fraction [tex]\(\frac{\sqrt{2} - 1}{1}\)[/tex]

1. Expression: [tex]\(\sqrt{2} - 1\)[/tex]
2. To rationalize the denominator:
- Multiply both the numerator and the denominator by the conjugate of [tex]\(\sqrt{2} - 1\)[/tex], which is [tex]\(\sqrt{2} + 1\)[/tex].

### Step 2: Multiply the Numerator and Denominator by the Conjugate

1. Numerator:
[tex]\[ (\sqrt{2} - 1) (\sqrt{2} + 1) \][/tex]

2. Denominator:
[tex]\[ (\sqrt{2} - 1) (\sqrt{2} + 1) \][/tex]

### Step 3: Simplify the Expression

Using the difference of squares formula, [tex]\((a - b)(a + b) = a^2 - b^2\)[/tex]:

1. Numerator:
[tex]\[ (\sqrt{2})^2 - (1)^2 = 2 - 1 = 1 \][/tex]
2. Denominator:
[tex]\[ (\sqrt{2})^2 - (1)^2 = 2 - 1 = 1 \][/tex]

### Step 4: Simplified Fraction

So, after rationalizing the denominator and simplifying, we get:

[tex]\[ \frac{(\sqrt{2} - 1)(\sqrt{2} + 1)}{(\sqrt{2} - 1)(\sqrt{2} + 1)} = \frac{1}{1} = 1 \][/tex]

Therefore, the simplified fraction after rationalizing the denominator is [tex]\(\boxed{1}\)[/tex].