Question

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Explain, in complete sentences, the effects of multiplying a radical or a complex number by its conjugate. Give an example of each.

Answer 1

Multiplying by the conjugate eliminates your radicals or imaginary parts (a special case, since \(i=\sqrt{-1}\)). It's just an application of the difference of squares:$$(a+b)(a-b)=a^2+ab-ab-b^2=a^2-b^2$$Notice that if either \(a,b\) had square roots or \(i\) terms, squaring would turn them into something nicer to look at. Hence why we use it for *rationalizing* a denominator.