Questions: Rationalize the denominator and simplify if possible. (sqrt(13)+17)/(sqrt(12)-8)

Solution Steps

To rationalize the denominator of the expression

\[ \frac{\sqrt{13}+17}{\sqrt{12}-8} \]

we multiply both the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{12}+8\):

\[ \frac{(\sqrt{13}+17)(\sqrt{12}+8)}{(\sqrt{12}-8)(\sqrt{12}+8)} \]

The denominator simplifies as follows:

\[ (\sqrt{12}-8)(\sqrt{12}+8) = (\sqrt{12})^2 - 8^2 = 12 - 64 = -52 \]

Now we expand the numerator:

\[ (\sqrt{13}+17)(\sqrt{12}+8) = \sqrt{13}\cdot\sqrt{12} + 8\sqrt{13} + 17\cdot\sqrt{12} + 17\cdot8 \]

This results in:

\[ \sqrt{156} + 8\sqrt{13} + 17\sqrt{12} + 136 \]

Putting it all together, we have:

\[ \frac{\sqrt{156} + 8\sqrt{13} + 17\sqrt{12} + 136}{-52} \]

The final expression simplifies to:

\[ -\frac{\sqrt{156}}{52} - \frac{8\sqrt{13}}{52} - \frac{17\sqrt{12}}{52} - \frac{136}{52} \]

This can be further simplified to:

\[ -\frac{34}{13} - \frac{17\sqrt{3}}{26} - \frac{2\sqrt{13}}{13} - \frac{\sqrt{39}}{26} \]

Thus, the final simplified result is:

\[ -\frac{34}{13} - \frac{17\sqrt{3}}{26} - \frac{2\sqrt{13}}{13} - \frac{\sqrt{39}}{26} \]

Final Answer