Answer
Hypotenuse $=2\sqrt{13}$ inches.
Work Step by Step
The given lengths of the legs are
$a=\sqrt {20}-\sqrt6$ inches
$b=\sqrt {20}+\sqrt6$ inches
By using Pythagorean Theorem.
Hypotenuse $h=\sqrt{a^2+b^2}$.
Substitute all values.
$\Rightarrow h=\sqrt{(\sqrt {20}-\sqrt6)^2+(\sqrt {20}+\sqrt6)^2}$
Use special formulas
$(A-B)^2=A^2-2AB+B^2$
and $(A+B)^2=A^2+2AB+B^2$
$\Rightarrow h=\sqrt{(\sqrt {20})^2-2(\sqrt {20})(\sqrt6)+(\sqrt6)^2+(\sqrt {20})^2+2(\sqrt {20})(\sqrt6)+(\sqrt6)^2}$
Simplify.
$\Rightarrow h=\sqrt{20-2\sqrt {120}+6+20+2\sqrt {120}+6}$
$\Rightarrow h=\sqrt{52}$
$\Rightarrow h=\sqrt{4\cdot 13}$
$\Rightarrow h=2\sqrt{13}$ inches.
