To produce an equivalent fraction with a rational denominator, we need to rationalize the denominator of the given fraction [tex]\(\frac{3}{\sqrt{17} - \sqrt{2}}\)[/tex].
Rationalizing the denominator involves multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(\sqrt{17} - \sqrt{2}\)[/tex] is [tex]\(\sqrt{17} + \sqrt{2}\)[/tex].
So, we multiply [tex]\(\frac{3}{\sqrt{17} - \sqrt{2}}\)[/tex] by [tex]\(\frac{\sqrt{17} + \sqrt{2}}{\sqrt{17} + \sqrt{2}}\)[/tex]:
[tex]\[
\frac{3}{\sqrt{17} - \sqrt{2}} \times \frac{\sqrt{17} + \sqrt{2}}{\sqrt{17} + \sqrt{2}} = \frac{3(\sqrt{17} + \sqrt{2})}{(\sqrt{17} - \sqrt{2})(\sqrt{17} + \sqrt{2})}
\][/tex]
When we multiply the denominators, we use the difference of squares formula:
[tex]\[
(\sqrt{17} - \sqrt{2})(\sqrt{17} + \sqrt{2}) = 17 - 2 = 15
\][/tex]
Thus, the expression becomes:
[tex]\[
\frac{3(\sqrt{17} + \sqrt{2})}{15}
\][/tex]
Which simplifies to:
[tex]\[
\frac{3\sqrt{17} + 3\sqrt{2}}{15}
\][/tex]
This shows that the original fraction's denominator has been rationalized. Hence, the fraction we need to multiply by to rationalize the denominator is:
[tex]\[
\boxed{\frac{\sqrt{17} + \sqrt{2}}{\sqrt{17} + \sqrt{2}}}
\][/tex]
