Question

Simplify. Rationalize the denominator.
–6/7+√5
-6 is numerater
7+√5 is denominater

Answer 1

To simplify the expression \(-\frac{6}{7 + \sqrt{5}}\) and rationalize the denominator, we will multiply both the numerator and denominator by the conjugate of the denominator, which is \(7 - \sqrt{5}\).

The expression becomes:

\[
-\frac{6}{7 + \sqrt{5}} \cdot \frac{7 - \sqrt{5}}{7 - \sqrt{5}} = -\frac{6(7 - \sqrt{5})}{(7 + \sqrt{5})(7 - \sqrt{5})}
\]

Now, simplify the denominator:

\[
(7 + \sqrt{5})(7 - \sqrt{5}) = 7^2 - (\sqrt{5})^2 = 49 - 5 = 44
\]

Next, simplify the numerator:

\[
-6(7 - \sqrt{5}) = -42 + 6\sqrt{5}
\]

Putting it all together, we have:

\[
-\frac{42 - 6\sqrt{5}}{44}
\]

This can be rewritten by separating the terms:

\[
-\frac{42}{44} + \frac{6\sqrt{5}}{44}
\]

We can simplify \(-\frac{42}{44}\) to \(-\frac{21}{22}\):

\[
-\frac{21}{22} + \frac{3\sqrt{5}}{22}
\]

Combining these terms, we write it as:

\[
\frac{-21 + 3\sqrt{5}}{22}
\]

Thus, the simplified form of \(-\frac{6}{7 + \sqrt{5}}\) is:

\[
\frac{-21 + 3\sqrt{5}}{22}
\]