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To rewrite the fraction \( \frac{3+\sqrt{2}}{1} \) using the difference of squares, we need to rationalize the denominator.
Here's how you can do it step by step:
1. Multiply the numerator and denominator by the conjugate of the denominator to rationalize the fraction.
\[ \frac{3+\sqrt{2}}{1} \times \frac{1}{1} = \frac{3+\sqrt{2}}{1} \times \frac{1-\sqrt{2}}{1-\sqrt{2}} \]
2. Multiply the numerators and denominators together to simplify.
\[ = \frac{(3+\sqrt{2})(1-\sqrt{2})}{1(1-\sqrt{2})} \]
3. Expand and simplify the numerator.
\[ = \frac{3 - 3\sqrt{2} + \sqrt{2} - 2}{1-\sqrt{2}} \]
\[ = \frac{1 - 2\sqrt{2}}{1-\sqrt{2}} \]
Therefore, the fraction \( \frac{3+\sqrt{2}}{1} \) when its denominator is rationalized becomes \( \frac{1 - 2\sqrt{2}}{1-\sqrt{2}} \).
