# Chapter 7 - Section 7.4 - Adding, Subtracting, and Multiplying Radical Expressions - Exercise Set - Page 439: 62

$10-22\sqrt{3}$

#### Work Step by Step

Using $(a+b)(c+d)=ac+ad+bc+bd$, the given expression, $\left( \sqrt{6}-4\sqrt{2} \right)\left( 3\sqrt{6}+\sqrt{2} \right) ,$ is equivalent to \begin{array}{l}\require{cancel} \sqrt{6}(3\sqrt{6})+\sqrt{6}(\sqrt{2})-4\sqrt{2}( 3\sqrt{6})-4\sqrt{2}(\sqrt{2}) .\end{array} Using the properties of radicals, the expression above simplifies to \begin{array}{l}\require{cancel} 3\sqrt{6(6)}+\sqrt{6(2)}-4(3)\sqrt{2(6)}-4\sqrt{2(2)} \\\\= 3\sqrt{36}+\sqrt{12}-12\sqrt{12}-4\sqrt{4} \\\\= 3\sqrt{36}+\sqrt{4\cdot3}-12\sqrt{4\cdot3}-4\sqrt{4} \\\\= 3\sqrt{(6)^2}+\sqrt{(2)^2\cdot3}-12\sqrt{(2)^2\cdot3}-4\sqrt{(2)^2} \\\\= 3(6)+2\sqrt{3}-12(2)\sqrt{3}-4(2) \\\\= 18+2\sqrt{3}-24\sqrt{3}-8 \\\\= (18-8)+(2\sqrt{3}-24\sqrt{3}) \\\\= 10-22\sqrt{3} .\end{array}

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