Question

to rationalize the denominator of \\(\frac{5 - \sqrt{7}}{9 - \sqrt{14}}\\), you should multiply the expression by which fraction?\
options: \\(\frac{\sqrt{14}}{\sqrt{14}}\\), \\(\frac{5 + \sqrt{7}}{9 - \sqrt{14}}\\), \\(\frac{9 - \sqrt{14}}{9 - \sqrt{14}}\\), \\(\frac{9 + \sqrt{14}}{9 + \sqrt{14}}\\)

Explanation:

Step1: Recall Rationalization Rule

To rationalize a denominator of the form \(a - \sqrt{b}\), we multiply by its conjugate \(a + \sqrt{b}\) (since \((x - y)(x + y)=x^2 - y^2\), which eliminates the square root in the denominator). Here, the denominator is \(9 - \sqrt{14}\), so its conjugate is \(9 + \sqrt{14}\). We need to multiply the fraction \(\frac{5 - \sqrt{7}}{9 - \sqrt{14}}\) by \(\frac{9 + \sqrt{14}}{9 + \sqrt{14}}\) (because this fraction is equal to 1, so it doesn't change the value of the original expression).

Step2: Identify the Correct Fraction

Looking at the options, the fraction \(\frac{9 + \sqrt{14}}{9 + \sqrt{14}}\) is the one that matches the conjugate of the denominator \(9 - \sqrt{14}\).

Answer:

The fraction \(\frac{9 + \sqrt{14}}{9 + \sqrt{14}}\) (the top - left option among the given boxes)