(b) \( \frac{1}{(3+2 \sqrt{2})} \)

To simplify \( \frac{1}{3 + 2 \sqrt{2}} \), you can multiply the numerator and the denominator by the conjugate of the denominator, \( 3 - 2 \sqrt{2} \). This will help eliminate the square root in the denominator. Thus, we get: \[ \frac{1 \cdot (3 - 2\sqrt{2})}{(3 + 2\sqrt{2})(3 - 2\sqrt{2})} = \frac{3 - 2\sqrt{2}}{3^2 - (2\sqrt{2})^2} = \frac{3 - 2\sqrt{2}}{9 - 8} = 3 - 2\sqrt{2}. \] So the simplified expression is \( 3 - 2\sqrt{2} \). The next time you encounter such expressions, remember that multiplying by the conjugate is a handy trick to simplify fractions with square roots! It’s not just about making it look nicer; you’re also making it easier to work with in equations or further calculations. To truly master rationalizing denominators, try practicing with various expressions. Grab a couple of examples with different combinations of terms, like \( \frac{1}{5 - \sqrt{3}} \) or \( \frac{1}{2 + \sqrt{7}} \) and see how they work out. The more you practice, the easier and quicker it becomes!