Chapter 10 - Exponents and Radicals - 10.5 Expressions Containing Several Radical Terms - 10.5 Exercise Set - Page 660: 31

$5\sqrt{2}+2$

Using $a(b+c)=ab+ac$ or the Distributive Property, the given expression, $ \sqrt{2}(5+\sqrt{2}) ,$ is equivalent to \begin{array}{l}\require{cancel} 5\sqrt{2}+\sqrt{2(2)} \\\\= 5\sqrt{2}+\sqrt{4} .\end{array} Using the properties of radicals, the expression above simplifies to \begin{array}{l}\require{cancel} 5\sqrt{2}+\sqrt{(2)^2} \\\\= 5\sqrt{2}+2 .\end{array}