The fraction which produce an equivalent fraction with a rational denominator is ![\left(\frac{\sqrt{17}+\sqrt{2}}{\sqrt{17}+\sqrt{2}}\right)](https://tex.z-dn.net/?f=%5Cleft%28%5Cfrac%7B%5Csqrt%7B17%7D%2B%5Csqrt%7B2%7D%7D%7B%5Csqrt%7B17%7D%2B%5Csqrt%7B2%7D%7D%5Cright%29)
Explanation:
The equation is ![\frac{3}{\sqrt{17}-\sqrt{2}}](https://tex.z-dn.net/?f=%5Cfrac%7B3%7D%7B%5Csqrt%7B17%7D-%5Csqrt%7B2%7D%7D)
To find the rational denominator, let us take conjugate of the denominator and multiply the conjugate with both numerator and denominator.
Rewriting the equation, we have,
![\frac{3}{\sqrt{17}-\sqrt{2}}\left(\frac{\sqrt{17}+\sqrt{2}}{\sqrt{17}+\sqrt{2}}\right)](https://tex.z-dn.net/?f=%5Cfrac%7B3%7D%7B%5Csqrt%7B17%7D-%5Csqrt%7B2%7D%7D%5Cleft%28%5Cfrac%7B%5Csqrt%7B17%7D%2B%5Csqrt%7B2%7D%7D%7B%5Csqrt%7B17%7D%2B%5Csqrt%7B2%7D%7D%5Cright%29)
Multiplying, we get,
![\frac{3(\sqrt{17}+\sqrt{2})}{(\sqrt{17})^{2}-(\sqrt{2})^{2}}](https://tex.z-dn.net/?f=%5Cfrac%7B3%28%5Csqrt%7B17%7D%2B%5Csqrt%7B2%7D%29%7D%7B%28%5Csqrt%7B17%7D%29%5E%7B2%7D-%28%5Csqrt%7B2%7D%29%5E%7B2%7D%7D)
Simplifying the denominator, we get,
![\frac{3(\sqrt{17}+\sqrt{2})}{17-2}](https://tex.z-dn.net/?f=%5Cfrac%7B3%28%5Csqrt%7B17%7D%2B%5Csqrt%7B2%7D%29%7D%7B17-2%7D)
Subtracting, the values of denominator,
![\frac{3(\sqrt{17}+\sqrt{2})}{15}](https://tex.z-dn.net/?f=%5Cfrac%7B3%28%5Csqrt%7B17%7D%2B%5Csqrt%7B2%7D%29%7D%7B15%7D)
Dividing the numerator and denominator,
![\frac{\sqrt{17}+\sqrt{2}}{5}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csqrt%7B17%7D%2B%5Csqrt%7B2%7D%7D%7B5%7D)
Hence, the denominator has become a rational denominator.
Thus, the fraction which produce an equivalent fraction with a rational denominator is ![\left(\frac{\sqrt{17}+\sqrt{2}}{\sqrt{17}+\sqrt{2}}\right)](https://tex.z-dn.net/?f=%5Cleft%28%5Cfrac%7B%5Csqrt%7B17%7D%2B%5Csqrt%7B2%7D%7D%7B%5Csqrt%7B17%7D%2B%5Csqrt%7B2%7D%7D%5Cright%29)