If `x=(sqrt(2)+1)/(sqrt(2)-1)` and `y=(sqrt(2)-1)/(sqrt(2)+1)` , what is the value of `(x+y)` ?

Options

A. `6`
B. `sqrt(2)`
C. `3`
D. `3sqrt(2)`

Answer

To find the value of \( x + y \) where \( x = \frac{\sqrt{2} + 1}{\sqrt{2} - 1} \) and \( y = \frac{\sqrt{2} - 1}{\sqrt{2} + 1} \), we will follow these steps:

### Step 1: Simplify \( x \)
We start with the expression for \( x \):
\[ x = \frac{\sqrt{2} + 1}{\sqrt{2} - 1} \]
To simplify this, we will rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, \( \sqrt{2} + 1 \):
\[ x = \frac{(\sqrt{2} + 1)(\sqrt{2} + 1)}{(\sqrt{2} - 1)(\sqrt{2} + 1)} \]

### Step 2: Calculate the Denominator
Using the difference of squares formula \( a^2 - b^2 \):
\[ (\sqrt{2} - 1)(\sqrt{2} + 1) = (\sqrt{2})^2 - (1)^2 = 2 - 1 = 1 \]

### Step 3: Calculate the Numerator
Now, calculate the numerator:
\[ (\sqrt{2} + 1)(\sqrt{2} + 1) = (\sqrt{2})^2 + 2(\sqrt{2})(1) + (1)^2 = 2 + 2\sqrt{2} + 1 = 3 + 2\sqrt{2} \]

### Step 4: Combine to Find \( x \)
Now, substituting back into the expression for \( x \):
\[ x = \frac{3 + 2\sqrt{2}}{1} = 3 + 2\sqrt{2} \]

### Step 5: Simplify \( y \)
Next, we simplify \( y \):
\[ y = \frac{\sqrt{2} - 1}{\sqrt{2} + 1} \]
Again, we rationalize the denominator by multiplying by the conjugate \( \sqrt{2} - 1 \):
\[ y = \frac{(\sqrt{2} - 1)(\sqrt{2} - 1)}{(\sqrt{2} + 1)(\sqrt{2} - 1)} \]

### Step 6: Calculate the Denominator for \( y \)
Using the difference of squares:
\[ (\sqrt{2} + 1)(\sqrt{2} - 1) = 2 - 1 = 1 \]

### Step 7: Calculate the Numerator for \( y \)
Now, calculate the numerator:
\[ (\sqrt{2} - 1)(\sqrt{2} - 1) = (\sqrt{2})^2 - 2(\sqrt{2})(1) + (1)^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2} \]

### Step 8: Combine to Find \( y \)
Thus, substituting back into the expression for \( y \):
\[ y = \frac{3 - 2\sqrt{2}}{1} = 3 - 2\sqrt{2} \]

### Step 9: Calculate \( x + y \)
Now we can find \( x + y \):
\[ x + y = (3 + 2\sqrt{2}) + (3 - 2\sqrt{2}) = 3 + 3 + 2\sqrt{2} - 2\sqrt{2} = 6 \]

### Final Answer
The value of \( x + y \) is:
\[ \boxed{6} \]