To rationalize the denominator of the fraction [tex]\(\frac{5 - \sqrt{7}}{9 - \sqrt{14}}\)[/tex], you need to eliminate the square root in the denominator. This is typically done by multiplying both the numerator and the denominator by the conjugate of the denominator.
The denominator of the given fraction is [tex]\(9 - \sqrt{14}\)[/tex]. The conjugate of [tex]\(9 - \sqrt{14}\)[/tex] is [tex]\(9 + \sqrt{14}\)[/tex].
Rationalizing the denominator involves multiplying the fraction by the fraction [tex]\(\frac{9 + \sqrt{14}}{9 + \sqrt{14}}\)[/tex], because multiplying by this form of 1 will not change the value of the original fraction.
Thus, the fraction we should use to rationalize the denominator [tex]\(9 - \sqrt{14}\)[/tex] is:
[tex]\[
\frac{9 + \sqrt{14}}{9 + \sqrt{14}}
\][/tex]
Therefore, the choice corresponding to the correct fraction to multiply by is:
[tex]\[
\boxed{\frac{9 + \sqrt{14}}{9 + \sqrt{14}}}
\][/tex]
This corresponds to the third option in the given list.
