Question P.6.13: Rationalizing a Numerator Rationalize the numerator: √x+h - ...
Rationalizing a Numerator
Rationalize the numerator:
\frac{\sqrt{x+h} – \sqrt{x}}{h}.
The conjugate of the numerator is \sqrt{x+h} +\sqrt{x}. If we multiply the numerator and denominator by \sqrt{x+h} +\sqrt{x}, the simplified numerator will not contain a radical. Therefore, we multiply by 1, choosing \frac{\sqrt{x+h} +\sqrt{x}}{\sqrt{x+h} +\sqrt{x}} for 1.
\frac{\sqrt{x+h} – \sqrt{x}}{h}=\frac{\sqrt{x+h} – \sqrt{x}}{h}·\frac{\sqrt{x+h} +\sqrt{x}}{\sqrt{x+h} +\sqrt{x}} Multiply by 1.
=\frac{(\sqrt{x+h})²-(\sqrt{x})²}{h(\sqrt{x+h}+\sqrt{x})} (\sqrt{a}-\sqrt{b})(\sqrt{a}+\sqrt{b}) = (\sqrt{a})^2 – (\sqrt{b})^2
=\frac{x + h – x}{h(\sqrt{x+h}+\sqrt{x})} (\sqrt{x+h})²=x+h and (\sqrt{x})²=x.
=\frac{h}{h(\sqrt{x+h}+\sqrt{x})} Simplify: x + h – x = h.
=\frac{1}{\sqrt{x+h}+\sqrt{x}}, h ≠ 0 Divide both the numerator and denominator by h.