following by rationalizing the denominator and reducing, if necessal \[ \frac{48}{5+\sqrt{13}} \]
Upstudy AI Solution
Answer
Solution
Answered by UpStudy AI and reviewed by a Professional Tutor
Explain 
Simplify this solution The Deep Dive
To rationalize the denominator of the expression \(\frac{48}{5+\sqrt{13}}\), we multiply the numerator and the denominator by the conjugate of the denominator, which is \(5 - \sqrt{13}\): \[ \frac{48(5 - \sqrt{13})}{(5 + \sqrt{13})(5 - \sqrt{13})} \] Calculating the denominator: \[ (5 + \sqrt{13})(5 - \sqrt{13}) = 5^2 - (\sqrt{13})^2 = 25 - 13 = 12 \] Now calculate the numerator: \[ 48(5 - \sqrt{13}) = 240 - 48\sqrt{13} \] Putting it all together, we have: \[ \frac{240 - 48\sqrt{13}}{12} \] Next, we can simplify this by dividing both terms in the numerator by 12: \[ \frac{240}{12} - \frac{48\sqrt{13}}{12} = 20 - 4\sqrt{13} \] Thus, the final simplified expression is: \[ 20 - 4\sqrt{13} \]
