2:
a: \(=\left(\sqrt{14}+\sqrt{6}\right)\sqrt{5-\sqrt{21}}\)
\(=\left(\sqrt{7}+\sqrt{3}\right)\cdot\sqrt{10-2\sqrt{21}}\)
\(=\left(\sqrt{7}+\sqrt{3}\right)\left(\sqrt{7}-\sqrt{3}\right)\)
=7-3=4
b: \(=\dfrac{\left(\sqrt{8-2\sqrt{15}}-\sqrt{8+2\sqrt{15}}\right)\left(\sqrt{4-2\sqrt{3}}+\sqrt{4+2\sqrt{3}}\right)}{2}\)
\(=\dfrac{\left(\sqrt{5}-\sqrt{3}-\sqrt{5}-\sqrt{3}\right)\left(\sqrt{3}-1+\sqrt{3}+1\right)}{2}\)
\(=-2\sqrt{3}\cdot\dfrac{2\sqrt{3}}{2}=-6\)
Bài 2:
\(\left(\dfrac{14}{\sqrt{14}}+\dfrac{\sqrt{12}+\sqrt{30}}{\sqrt{2}+\sqrt{5}}\right)\sqrt{5-\sqrt{21}}\)
\(=\left(\dfrac{\sqrt{14}\cdot\sqrt{14}}{\sqrt{14}}+\dfrac{\sqrt{6}\cdot\left(\sqrt{2}+\sqrt{5}\right)}{\sqrt{2}+\sqrt{5}}\right)\sqrt{5-\sqrt{21}}\)
\(=\left(\sqrt{14}+\sqrt{6}\right)\sqrt{5-\sqrt{21}}\)
\(=\sqrt{2}\left(\sqrt{7}+\sqrt{3}\right)\sqrt{5-\sqrt{21}}\)
\(=\left(\sqrt{7}+\sqrt{3}\right)\sqrt{2\cdot\left(5-\sqrt{21}\right)}\)
\(=\left(\sqrt{7}+\sqrt{3}\right)\sqrt{10-2\sqrt{21}}\)
\(=\left(\sqrt{7}+\sqrt{3}\right)\sqrt{\left(\sqrt{7}\right)^2-2\cdot\sqrt{7}\cdot\sqrt{3}+\left(\sqrt{3}\right)^2}\)
\(=\left(\sqrt{7}+\sqrt{3}\right)\sqrt{\left(\sqrt{7}-\sqrt{3}\right)^2}\)
\(=\left(\sqrt{7}+\sqrt{3}\right)\left(\sqrt{7}-\sqrt{3}\right)\)
\(=\left(\sqrt{7}\right)^2-\left(\sqrt{3}\right)^2\)
\(=7-3\)
\(=4\)
b) \(\left(\sqrt{4-\sqrt{15}}-\sqrt{4+\sqrt{15}}\right)\left(\sqrt{2-\sqrt{3}}+\sqrt{2+\sqrt{3}}\right)\)
\(=\dfrac{2\cdot\left(\sqrt{4-\sqrt{15}}-\sqrt{4+\sqrt{15}}\right)\left(\sqrt{2-\sqrt{3}}+\sqrt{2+\sqrt{3}}\right)}{2}\)
\(=\dfrac{\left(\sqrt{8-2\sqrt{15}}-\sqrt{8+2\sqrt{15}}\right)\left(\sqrt{4-2\sqrt{3}}+\sqrt{4+2\sqrt{3}}\right)}{2}\)
\(=\dfrac{\left(\sqrt{\left(\sqrt{5}\right)^2-2\cdot\sqrt{5}\cdot\sqrt{3}+\left(\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{5}\right)^2+2\sqrt{5}\cdot\sqrt{3}+\left(\sqrt{3}\right)^2}\right)\left(\sqrt{\left(\sqrt{3}\right)^2-2\cdot\sqrt{3}\cdot1+1^2}+\sqrt{\left(\sqrt{3}\right)^2+2\cdot\sqrt{3}\cdot1+1^2}\right)}{2}\)
\(=\dfrac{\left[\sqrt{\left(\sqrt{5}-\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{5}+\sqrt{3}\right)^2}\right]\left[\sqrt{\left(\sqrt{3}-1\right)^2}+\sqrt{\left(\sqrt{3}+1\right)^2}\right]}{2}\)
\(=\dfrac{\left(\sqrt{5}-\sqrt{3}-\sqrt{5}-\sqrt{3}\right)\left(\sqrt{3}-1+\sqrt{3}+1\right)}{2}\)
\(=\dfrac{-2\sqrt{3}\cdot2\sqrt{3}}{2}\)
\(=-\sqrt{3}\cdot2\sqrt{3}\)
\(=-2\cdot3\)
\(=-6\)
\(a)\sqrt{2x+2\sqrt{x^2-4}}\)
\(=\sqrt{x-2+2\sqrt{\left(x-2\right)\left(x+2\right)}+x+2}\)
\(=\sqrt{\left(\sqrt{x+2}+\sqrt{x-2}\right)^2}=\left|\sqrt{x+2}+\sqrt{x-2}\right|=\sqrt{x+2}+\sqrt{x-2}\) ( vì x>2)
\(b)\sqrt{4x+2\sqrt{4x^2-1}}\)
\(=\sqrt{2x-1+\sqrt{\left(2x-1\right)\left(2x+1\right)+2x+1}}\)
\(=\sqrt{\left(\sqrt{2x-1}+\sqrt{2x+1}\right)^2}\)
\(=\left|\sqrt{2x-1}+\sqrt{2x+1}\right|\)
\(=\sqrt{2x-1}+\sqrt{2x+1}\) (Vì x>1/2)
\(c)\sqrt{4+2\sqrt{4x-x^2}}\)
\(=\sqrt{x+2\sqrt{x\left(4-x\right)}+4-x}\)
\(=\sqrt{\left(\sqrt{x}+\sqrt{4-x}\right)^2}\)
\(=\left|\sqrt{x}+\sqrt{4-x}\right|=\sqrt{x}+\sqrt{4-x}\) ( vì 0<x<1)
