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].
