What is the reason behind multiplying by conjugates?

I am currently studying single variable calculus and throughout the lessons from the text I'm using, the reasoning as to why one would multiply by conjugates is merely shown as a technique to be used.

I see one reason has to do with rationalizing denominators. But there were some problems I encountered which involved the multiplication by conjugates without the rationalization of denominators, for instance, in evaluating $\displaystyle\lim_{x_\to\infty}(\sqrt{x^2+ax}-\sqrt{x^2+bx})$ or showing that the function $ln(x+\sqrt{x^2+1})$ is odd.

If anyone can enlighten me here it would be much appreciated.

  • $\begingroup$ What do you mean by reason? It's an algebraic tool to manipulate and reach certain points. $\endgroup$ – Aaron Maroja Dec 16 '14 at 20:29

When you multiply by the conjugate over the conjugate, you're exploiting the rule $(a-b)(a+b)=a^2-b^2$. The latter permits you to remove the radicals in the numerator and to get a form which is simpler to manipulate than the original.

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