Section C 26. If \( \frac{1+\sqrt{2}}{1-\sqrt{2}}+\frac{1-\sqrt{2}}{1-\sqrt{2}}=a+b \sqrt{2} \), Find the value of a and b

To simplify the expression \( \frac{1+\sqrt{2}}{1-\sqrt{2}}+\frac{1-\sqrt{2}}{1-\sqrt{2}} \), we first deal with each term separately. **Step 1: Simplify the first term** We can rationalize the first term: \[ \frac{1+\sqrt{2}}{1-\sqrt{2}} \cdot \frac{1+\sqrt{2}}{1+\sqrt{2}} = \frac{(1+\sqrt{2})^2}{(1-\sqrt{2})(1+\sqrt{2})} \] Calculating the denominator: \[ (1-\sqrt{2})(1+\sqrt{2}) = 1^2 - (\sqrt{2})^2 = 1 - 2 = -1 \] Now calculating the numerator: \[ (1+\sqrt{2})^2 = 1^2 + 2 \cdot 1 \cdot \sqrt{2} + (\sqrt{2})^2 = 1 + 2\sqrt{2} + 2 = 3 + 2\sqrt{2} \] Now, substituting back into the simplification gives us: \[ \frac{(1+\sqrt{2})^2}{(1-\sqrt{2})(1+\sqrt{2})} = \frac{3 + 2\sqrt{2}}{-1} = -3 - 2\sqrt{2} \] **Step 2: Simplify the second term** The second term is simpler: \[ \frac{1-\sqrt{2}}{1-\sqrt{2}} = 1 \] **Step 3: Combine both parts** Now, we can combine the simplified terms: \[ -3 - 2\sqrt{2} + 1 = -2 - 2\sqrt{2} \] This can be rewritten as: \[ (-2) + (-2)\sqrt{2} \] So, we have \( a = -2 \) and \( b = -2 \). Thus, the final values are: \[ \boxed{(-2, -2)} \]