To rationalize the denominator of the expression [tex]\(\frac{5 - \sqrt{7}}{9 - \sqrt{14}}\)[/tex], you need to eliminate the square root from the denominator. This is achieved by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial [tex]\(a - b\)[/tex] is [tex]\(a + b\)[/tex].
For the denominator [tex]\(9 - \sqrt{14}\)[/tex], its conjugate is [tex]\(9 + \sqrt{14}\)[/tex].
Thus, we need to multiply the fraction by [tex]\(\frac{9 + \sqrt{14}}{9 + \sqrt{14}}\)[/tex], which is the conjugate of [tex]\(9 - \sqrt{14}\)[/tex].
So the correct fraction to multiply by is:
[tex]\[
\frac{9 + \sqrt{14}}{9 + \sqrt{14}}
\][/tex]
Hence, the answer is:
[tex]\[
\frac{9+\sqrt{14}}{9+\sqrt{14}}
\][/tex]
