To rationalize the denominator of [tex]\frac{5-\sqrt{7}}{9-\sqrt{14}}[/tex], you should multiply the expression by which fraction?

A. [tex]\frac{5+\sqrt{7}}{9-\sqrt{14}}[/tex]
B. [tex]\frac{9-\sqrt{14}}{9-\sqrt{14}}[/tex]
C. [tex]\frac{9+\sqrt{14}}{9+\sqrt{14}}[/tex]
D. [tex]\frac{\sqrt{14}}{\sqrt{14}}[/tex]



Answer :

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]