Find the value of m and n : if :
` ( 3+ sqrt2 )/( 3- sqrt2) = m + n sqrt2 `

Answer

To solve the equation \(\frac{3 + \sqrt{2}}{3 - \sqrt{2}} = m + n \sqrt{2}\), we will follow these steps:

### Step 1: Rationalize the Denominator
We start with the left-hand side (LHS):
\[ \frac{3 + \sqrt{2}}{3 - \sqrt{2}} \]
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(3 + \sqrt{2}\):
\[ \frac{(3 + \sqrt{2})(3 + \sqrt{2})}{(3 - \sqrt{2})(3 + \sqrt{2})} \]

### Step 2: Simplify the Denominator
Using the difference of squares formula, the denominator simplifies as follows:
\[ (3 - \sqrt{2})(3 + \sqrt{2}) = 3^2 - (\sqrt{2})^2 = 9 - 2 = 7 \]

### Step 3: Expand the Numerator
Now, we expand the numerator:
\[ (3 + \sqrt{2})(3 + \sqrt{2}) = 3^2 + 2 \cdot 3 \cdot \sqrt{2} + (\sqrt{2})^2 = 9 + 6\sqrt{2} + 2 = 11 + 6\sqrt{2} \]

### Step 4: Combine the Results
Now, we can combine the results from the numerator and the denominator:
\[ \frac{11 + 6\sqrt{2}}{7} \]
This can be separated into two fractions:
\[ \frac{11}{7} + \frac{6\sqrt{2}}{7} \]

### Step 5: Compare with the Right-Hand Side
Now we have:
\[ \frac{11}{7} + \frac{6\sqrt{2}}{7} = m + n\sqrt{2} \]
From this, we can compare coefficients:
- The constant term gives us \(m = \frac{11}{7}\)
- The coefficient of \(\sqrt{2}\) gives us \(n = \frac{6}{7}\)

### Final Answer
Thus, the values of \(m\) and \(n\) are:
\[ m = \frac{11}{7}, \quad n = \frac{6}{7} \]

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