If `x=(1)/(sqrt(2)+1)` then `(x+1)` equal to

Options

A. `2`
B. `sqrt(2)`
C. `sqrt(2)+1`
D. `sqrt(2)-1`

Answer

To solve the problem where \( x = \frac{1}{\sqrt{2} + 1} \) and we need to find \( x + 1 \), we can follow these steps:

### Step 1: Rationalize the denominator
We start with the expression for \( x \):
\[ x = \frac{1}{\sqrt{2} + 1} \]
To simplify this, we can multiply the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{2} - 1 \):
\[ x = \frac{1 \cdot (\sqrt{2} - 1)}{(\sqrt{2} + 1)(\sqrt{2} - 1)} \]

### Step 2: Apply the difference of squares
Now, we simplify the denominator using the difference of squares formula \( (a + b)(a - b) = a^2 - b^2 \):
\[ (\sqrt{2} + 1)(\sqrt{2} - 1) = (\sqrt{2})^2 - (1)^2 = 2 - 1 = 1 \]
Thus, we have:
\[ x = \sqrt{2} - 1 \]

### Step 3: Calculate \( x + 1 \)
Now we need to find \( x + 1 \):
\[ x + 1 = (\sqrt{2} - 1) + 1 \]
This simplifies to:
\[ x + 1 = \sqrt{2} \]

### Conclusion
Therefore, the value of \( x + 1 \) is:
\[ \sqrt{2} \]

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