## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$\dfrac{20}{\sqrt{10}+\sqrt{15}}$
Multiplying by an expression equal to $1$ which will make the numerator a perfect power of the index, then the rationalized-numerator form of the given expression is \begin{array}{l}\require{cancel} \dfrac{4\sqrt{5}}{\sqrt{2}+\sqrt{3}} \\\\= \dfrac{4\sqrt{5}}{\sqrt{2}+\sqrt{3}}\cdot\dfrac{4\sqrt{5}}{4\sqrt{5}} \\\\= \dfrac{(4\sqrt{5})^2}{4\sqrt{5}(\sqrt{2}+\sqrt{3})} \\\\= \dfrac{16(5)}{4\sqrt{5}(\sqrt{2}+\sqrt{3})} \\\\= \dfrac{80}{4\sqrt{5}(\sqrt{2}+\sqrt{3})} .\end{array} Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{80}{4\sqrt{5}(\sqrt{2}+\sqrt{3})} \\\\= \dfrac{80}{4\sqrt{5}(\sqrt{2})+4\sqrt{5}(\sqrt{3})} \\\\= \dfrac{80}{4\sqrt{10}+4\sqrt{15}} \\\\= \dfrac{\cancel4^{20}}{\cancel4\sqrt{10}+\cancel4\sqrt{15}} \\\\= \dfrac{20}{\sqrt{10}+\sqrt{15}} .\end{array}