Rationalizing Expressions (C)

The formula for summing a geometric progression can be used to rewrite expressions containing the difference of two roots of the same order.

For example, we have for $a \geq 0$ and $b\geq 0$

$$a-b = (\sqrt{a}-\sqrt{b})(\sqrt{a} + \sqrt{b})$$

so, for example,

$$\begin{array} {rcl} \displaystyle{\frac{\sqrt{x+h}-\sqrt{x}}... ... & = & \displaystyle{\frac{1}{\sqrt{x+h}+\sqrt{x}}}.\end{array}$$

Another example is

$$a-b = (\sqrt[3]{a}-\sqrt[3]{b})\times((\sqrt[3]{a})^2 + \sqrt[3]{a}\sqrt[3]{b} + (\sqrt[3]{b})^2)$$

so

$$\begin{array} {rcl} \displaystyle{\frac{\sqrt[3]{x+h}-\sqrt[... ...]{x+h})^2+\sqrt[3]{x+h}\sqrt[3]{x}+(\sqrt[3]{x})^2}}\end{array}$$