Identifying the Values of a and b in the Given Equation

Given Equation: \(\frac{3+\sqrt{7}}{3-\sqrt{2}} = a + b\sqrt{2}\)

Step 1: Rationalize the Denominator
To simplify the given equation, we need to rationalize the denominator. Multiply the numerator and the denominator by the conjugate of the denominator:
\[\frac{3+\sqrt{7}}{3-\sqrt{2}} \times \frac{3+\sqrt{2}}{3+\sqrt{2}} = a + b\sqrt{2}\]
This simplifies to:
\[\frac{3(3) + 3\sqrt{2} + 3\sqrt{7} + \sqrt{14}}{3(3) - 3\sqrt{2} + 3\sqrt{2} - 2} = a + b\sqrt{2}\]

Step 2: Simplify the Numerator
Simplify the numerator:
\[\frac{9 + 3\sqrt{2} + 3\sqrt{7} + \sqrt{14}}{9 - 2} = a + b\sqrt{2}\]
\[\frac{9 + 3\sqrt{2} + 3\sqrt{7} + \sqrt{14}}{7} = a + b\sqrt{2}\]

Step 3: Compare Coefficients
Now, compare the coefficients of \(\sqrt{2}\) on both sides of the equation:
Coefficient of \(\sqrt{2}\) on the left side = \( \frac{3\sqrt{2}}{7} + \frac{\sqrt{14}}{7} \)
Coefficient of \(\sqrt{2}\) on the right side = \( b \)

Step 4: Solve for a and b
Equating the coefficients:
\( \frac{3}{7} = 0 \) and \( \frac{1}{7} = b \)
Therefore, the values of a and b are:
\( a = 0 \) and \( b = \frac{1}{7} \)