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The given expression is:

\[
\sqrt{2} + 1 - \frac{1}{\sqrt{2} + 1}
\]

To simplify this, we'll rationalize the denominator in the fraction part.

### Step 1: Rationalize the denominator

The fraction part is:

\[
\frac{1}{\sqrt{2} + 1}
\]

To rationalize, multiply both the numerator and denominator by \( \sqrt{2} - 1 \):

\[
\frac{1}{\sqrt{2} + 1} \times \frac{\sqrt{2} - 1}{\sqrt{2} - 1} = \frac{\sqrt{2} - 1}{(\sqrt{2})^2 - (1)^2} = \frac{\sqrt{2} - 1}{2 - 1} = \sqrt{2} - 1
\]

So, the expression becomes:

\[
\sqrt{2} + 1 - (\sqrt{2} - 1)
\]

### Step 2: Simplify the expression

Now, simplify:

\[
\sqrt{2} + 1 - \sqrt{2} + 1 = 2
\]

Thus, the simplified expression is \( 2 \).

Would you like further clarification or have any questions?

Here are five related problems:
1. Simplify \( \sqrt{3} + 1 - \frac{1}{\sqrt{3} + 1} \).
2. Rationalize \( \frac{1}{\sqrt{5} + 2} \).
3. Expand \( (1 + \sqrt{2})^2 \).
4. Simplify \( \frac{3}{\sqrt{3} - 1} \).
5. Simplify \( 1 - \frac{1}{\sqrt{2} + 1} \).

**Tip**: Rationalizing the denominator is a common technique when dealing with square roots in the denominator. Multiply by the conjugate to simplify the expression.

Learn how to simplify the expression √2 + 1 − 1 / (√2 + 1) through rationalization of the denominator. This problem covers key algebraic concepts, including rationalization and expression simplification, suitable for Grades 9-11.

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