If the product of two surds is a rational number, then they are said to be rationalization factor to each other, for example \(9\sqrt 7 \) and \(5\sqrt 7 \) are rationalized to each other as \(9\sqrt 7  \times 5\sqrt 7  = 315\), a rational number. Similarly \(\sqrt 5  + \sqrt 3 \) and \(\sqrt 5  - \sqrt 3 \) are rationalizing factor of each other.

Again \(\left( {{5^{1/3}} + {3^{1/3}}} \right)\) and \(\left( {{5^{2/3}} + {3^{2/3}} - {{15}^{1/3}}} \right)\) are rationalizing factor of each other.

Example 01: Find the simplified value of \(\frac{1}{{\sqrt {14}  + \sqrt 9 }} + \frac{1}{{\sqrt {19}  + \sqrt {14} }} + .... + \frac{1}{{\sqrt {324}  + \sqrt {319} }}\)

Example 02: If \(\frac{1}{{\sqrt[3]{7} + \sqrt[3]{3}}} = a\sqrt[3]{{49}} + b\sqrt[3]{9} + c\sqrt[3]{{21}}\), find the value of \(a,\;b\) and \(c\)