Answer
$17+31\sqrt{2}$
Work Step by Step
Distributing using the FOIL method gives:
$=1\cdot5+1\cdot\sqrt{2}+5\sqrt{72}+\sqrt{72}\cdot\sqrt{2}$
$=5+\sqrt{2}+5\sqrt{72}+\sqrt{72\cdot }\sqrt{2}$
$=5+\sqrt{2}+5\sqrt{36\cdot2}+\sqrt{36\cdot2}\cdot \sqrt{2}$
Recall the property (pg. 367):
$\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{ab}$ (if $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers)
Applying this property, we get:
$5+\sqrt{2}+5\sqrt{36\cdot2}+\sqrt{36\cdot2}\cdot \sqrt{2}$
$=5+\sqrt{2}+5\sqrt{36}\cdot\sqrt{2}+\sqrt{36}\cdot\sqrt{2}\sqrt{2}$
Recall that $6^2=36$ and $2^2=4$,
Thus, the expression above is equivalent to:
$=5+\sqrt{2}+5\cdot6\sqrt{2}+6\sqrt{2\cdot2}$
$=5+1\sqrt{2}+30\sqrt{2}+6\sqrt{4}$
$=5+1\sqrt{2}+30\sqrt{2}+6\cdot2$
$=5+1\sqrt{2}+30\sqrt{2}+12$
$=17+1\sqrt{2}+30\sqrt{2}$
$=17+(1+30)\sqrt{2}$
$=17+31\sqrt{2}$
