Question

Simplify. Rationalize the denominator.
7/√7–√11
7 is numerator
√7-√11 is denominater

Answer 1

To simplify and rationalize the expression \(\frac{7}{\sqrt{7} - \sqrt{11}}\), we can multiply the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{7} + \sqrt{11}\).

Let's proceed with the calculation:

\[
\frac{7}{\sqrt{7} - \sqrt{11}} \cdot \frac{\sqrt{7} + \sqrt{11}}{\sqrt{7} + \sqrt{11}} = \frac{7(\sqrt{7} + \sqrt{11})}{(\sqrt{7} - \sqrt{11})(\sqrt{7} + \sqrt{11})}
\]

Now, calculate the denominator:

\[
(\sqrt{7} - \sqrt{11})(\sqrt{7} + \sqrt{11}) = \sqrt{7}^2 - \sqrt{11}^2 = 7 - 11 = -4
\]

Now, plug this back into the expression:

\[
\frac{7(\sqrt{7} + \sqrt{11})}{-4}
\]

This simplifies to:

\[
-\frac{7(\sqrt{7} + \sqrt{11})}{4}
\]

So the final simplified form of the expression \(\frac{7}{\sqrt{7} - \sqrt{11}}\), with a rationalized denominator, is:

\[
-\frac{7(\sqrt{7} + \sqrt{11})}{4}
\]