\(\sqrt{\frac{2}{2-\sqrt{3}}}-\sqrt{\frac{2}{2+\sqrt{3}}}\)
\(=\sqrt{\frac{2\left(2+\sqrt{3}\right)}{4-3}}-\sqrt{\frac{2\left(2-\sqrt{3}\right)}{4-3}}\)
\(=\sqrt{4+2\sqrt{3}}-\sqrt{4-2\sqrt{3}}\)
\(=\sqrt{\left(\sqrt{3}+1\right)^2}-\sqrt{\left(\sqrt{3}-1\right)^2}\)
\(=\left|\sqrt{3}+1\right|-\left|\sqrt{3}-1\right|\)
\(=\sqrt{3}+1-\sqrt{3}+1\)( \(\sqrt{3}+1>0\) và \(\sqrt{3}-1>0\) )
\(=2\)
\(\)
\(\sqrt{2\left(2+\sqrt{3}\right)}-\sqrt{2\left(2-\sqrt{3}\right)}\))
\(\sqrt{4+2\sqrt{3}}-\sqrt{4-2\sqrt{3}}\)
\(\sqrt{\left(1+\sqrt{3}\right)^2}-\sqrt{\left(1-\sqrt{3}\right)^2}\)
\(1+\sqrt{3}-\sqrt{3}+1\)
\(2\)
\(P=\sqrt{\frac{2}{2-\sqrt{3}}}-\sqrt{\frac{2}{2+\sqrt{3}}}\)
\(P=\sqrt{\frac{2\left(2+\sqrt{3}\right)}{\left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right)}}-\sqrt{\frac{2\left(2-\sqrt{3}\right)}{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}}\)
\(P=\sqrt{\frac{4+2\sqrt{3}}{4-3}}-\sqrt{\frac{4-2\sqrt{3}}{4-3}}\)
\(P=\sqrt{4+2\sqrt{3}}-\sqrt{4-2\sqrt{3}}\)
\(P^2=\left(\sqrt{4+2\sqrt{3}}-\sqrt{4-2\sqrt{3}}\right)^2\)
\(P^2=4+2\sqrt{3}-2\sqrt{\left(4+2\sqrt{3}\right)\left(4-2\sqrt{3}\right)}+4-2\sqrt{3}\)
\(P^2=8-2\sqrt{16-4.3}=8-2\sqrt{4}=8-4=4\)
\(\Rightarrow P=\pm2\).
