Answer
$63-38\sqrt{2}$
Work Step by Step
Using the FOIL Method which is given by $(a+b)(c+d)=ac+ad+bc+bd,$ the given expression, $
(3-4\sqrt{2})(5-6\sqrt{2})
,$ is equivalent to
\begin{align*}
&
3(5)+3(-6\sqrt{2})-4\sqrt{2}(5)-4\sqrt{2}(-6\sqrt{2})
\\&=
15-18\sqrt{2}-20\sqrt{2}+24(2)
\\&=
15-18\sqrt{2}-20\sqrt{2}+48
\\&=
(15+48)+(-18\sqrt{2}-20\sqrt{2})
&\left( \text{combine like terms} \right)
\\&=
\\&=
63-38\sqrt{2}
.\end{align*}
Hence, the product of the given expression is $
63-38\sqrt{2}
$.
