Unit Test Unit Test Active 4 11 12 13 14 15 16 TIME REMAINING 01:03:43 Multiplying \( \frac{3}{\sqrt{17}-\sqrt{2}} \) by which fraction will produce an equivalent fraction with a rational denominator? \( \frac{\sqrt{17}-\sqrt{2}}{\sqrt{17}} \)

To rationalize the denominator of the expression \( \frac{3}{\sqrt{17} - \sqrt{2}} \), you should multiply both the numerator and the denominator by the **conjugate** of the denominator. The conjugate of \( \sqrt{17} - \sqrt{2} \) is \( \sqrt{17} + \sqrt{2} \). Here's how it works step-by-step: 1. **Identify the Conjugate:** \[ \text{Conjugate of } (\sqrt{17} - \sqrt{2}) \text{ is } (\sqrt{17} + \sqrt{2}) \] 2. **Multiply the Original Fraction by the Conjugate Over Itself:** \[ \frac{3}{\sqrt{17} - \sqrt{2}} \times \frac{\sqrt{17} + \sqrt{2}}{\sqrt{17} + \sqrt{2}} = \frac{3(\sqrt{17} + \sqrt{2})}{(\sqrt{17})^2 - (\sqrt{2})^2} \] 3. **Simplify the Denominator Using the Difference of Squares:** \[ (\sqrt{17})^2 - (\sqrt{2})^2 = 17 - 2 = 15 \] 4. **Final Rationalized Form:** \[ \frac{3(\sqrt{17} + \sqrt{2})}{15} = \frac{\sqrt{17} + \sqrt{2}}{5} \] **Therefore, the fraction you should multiply by is:** \[ \frac{\sqrt{17} + \sqrt{2}}{\sqrt{17} + \sqrt{2}} \] This multiplication will result in an equivalent fraction with a rational denominator. **Final Answer:** Multiply by the conjugate fraction \(\dfrac{\sqrt{17} + \sqrt{2}}{\sqrt{17} + \sqrt{2}}\).