Solution 3.1:3a
From Förberedande kurs i matematik 1
First expand the expression
\displaystyle \begin{align}
& \left( \sqrt{5}-\sqrt{2} \right)\left( \sqrt{5}3\sqrt{2} \right)=\sqrt{5}\centerdot \sqrt{5}+\sqrt{5}\centerdot \sqrt{2}-\sqrt{2}\centerdot \sqrt{5}-\sqrt{2}\centerdot \sqrt{2} \\
& =\sqrt{5}\centerdot \sqrt{5}-\sqrt{2}\centerdot \sqrt{2} \\
\end{align}
Because
\displaystyle \sqrt{5}
and
\displaystyle \sqrt{2}
are defined as those numbers which, when multiplied with themselves give
\displaystyle \text{5}
and
\displaystyle 2 respectively,
\displaystyle \sqrt{5}\centerdot \sqrt{5}-\sqrt{2}\centerdot \sqrt{2}=5-2=3
NOTE: The expansion of
\displaystyle \left( \sqrt{5}-\sqrt{2} \right)\left( \sqrt{5}3\sqrt{2} \right)
can also be done directly with the conjugate rule
\displaystyle \left( a-b \right)(a+b)=a^{\text{2}}-b^{\text{2}}
using
\displaystyle a=\sqrt{5}
and
\displaystyle b=\sqrt{2}.
