Question

Simplify. Rationalize the denominator. - radical 3 / 8 + radical 3

Answer 1

To simplify and rationalize the expression \(-\frac{\sqrt{3}}{8 + \sqrt{3}}\), we will first multiply the numerator and the denominator by the conjugate of the denominator.

The conjugate of \(8 + \sqrt{3}\) is \(8 - \sqrt{3}\).

Now, let's perform the multiplication:

\[
-\frac{\sqrt{3}}{8 + \sqrt{3}} \cdot \frac{8 - \sqrt{3}}{8 - \sqrt{3}} = -\frac{\sqrt{3} (8 - \sqrt{3})}{(8 + \sqrt{3})(8 - \sqrt{3})}
\]

Now, calculate the denominator using the difference of squares:

\[
(8 + \sqrt{3})(8 - \sqrt{3}) = 8^2 - (\sqrt{3})^2 = 64 - 3 = 61
\]

Next, calculate the numerator:

\[
-\sqrt{3} (8 - \sqrt{3}) = -8\sqrt{3} + 3
\]

Putting it all together, we have:

\[
-\frac{8\sqrt{3} - 3}{61} = \frac{3 - 8\sqrt{3}}{61}
\]

So the simplified and rationalized expression is:

\[
\frac{3 - 8\sqrt{3}}{61}
\]