Atrisinājums-1
Veicam ekvivalentus pārveidojumus:
\[\begin{aligned}
\sqrt{4+2 \sqrt{3}} & -\sqrt{4-2 \sqrt{3}}=\sqrt{(\sqrt{3})^{2}+2 \cdot \sqrt{3} \cdot 1+1^{2}}-\sqrt{(\sqrt{3})^{2}-2 \cdot \sqrt{3} \cdot 1+1^{2}}= \\
& =\sqrt{(\sqrt{3}+1)^{2}}-\sqrt{(\sqrt{3}-1)^{2}}=\sqrt{3}+1-(\sqrt{3}-1)=2
\end{aligned}\]
Līdz ar to esam ieguvuši, ka dotais skaitlis ir racionāls skaitlis.
Solution 1
Perform equivalent transformations:
\[\begin{aligned}
\sqrt{4+2 \sqrt{3}} & -\sqrt{4-2 \sqrt{3}}=\sqrt{(\sqrt{3})^{2}+2 \cdot \sqrt{3} \cdot 1+1^{2}}-\sqrt{(\sqrt{3})^{2}-2 \cdot \sqrt{3} \cdot 1+1^{2}}= \\
& =\sqrt{(\sqrt{3}+1)^{2}}-\sqrt{(\sqrt{3}-1)^{2}}=\sqrt{3}+1-(\sqrt{3}-1)=2
\end{aligned}\]
Thus we have obtained that the given number is a rational number.
## Solution 2
By squaring the given number, we obtain
\[\left( \sqrt{4+2 \sqrt{3}}-\sqrt{4-2 \sqrt{3}} \right)^{2} =
4 + 2\sqrt{3} - 2 \sqrt{ \left( 4+2 \sqrt{3} \right) \left( 4-2 \sqrt{3} \right)}
+ 4 - 2\sqrt{3}=8-2 \sqrt{16-12}=4.\]
Since the square of the number is
\(4\), the given number is rational.