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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.