Section 1.3 : Radicals
18. Rationalize the denominator. Assume that \(x\) and \(y\) are positive.
\[\frac{4}{{\sqrt x + 2\sqrt y }}\] Show SolutionFor this problem we need to multiply the numerator and denominator by \(\sqrt x - 2\sqrt y \) in order to rationalize the denominator.
\[\frac{4}{{\sqrt x + 2\sqrt y }} = \frac{4}{{\sqrt x + 2\sqrt y }}\frac{{\sqrt x - 2\sqrt y }}{{\sqrt x - 2\sqrt y }} = \frac{{4\left( {\sqrt x - 2\sqrt y } \right)}}{{\left( {\sqrt x + 2\sqrt y } \right)\left( {\sqrt x - 2\sqrt y } \right)}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{4\sqrt x - 8\sqrt y }}{{x - 4y}}}}\]