Surds - Applications of Surds

A cone has a volume of \(\left( 9\sqrt{5}+13\sqrt{7} \right)\pi \)cm\(^3\) and a radius of \(\left( \sqrt{5}+\sqrt{7} \right)\)cm. Find the height of the cone in the form \(\frac{a\sqrt{7}+b\sqrt{5}}{2}\), where \(a\) and \(b\) are integers.