a) \(\sqrt{4-\sqrt{7}}-\sqrt{4+\sqrt{7}}\)
\(=\sqrt{\dfrac{1}{2}}\left(\sqrt{8-2\sqrt{7}}-\sqrt{8+2\sqrt{7}}\right)\)\(=\sqrt{\dfrac{1}{2}}\left(\sqrt{\left(\sqrt{7}-1\right)^2}-\sqrt{\left(\sqrt{7}+1\right)^2}\right)\)
\(=\sqrt{\dfrac{1}{2}}\left(\left|\sqrt{7}-1\right|-\sqrt{7}-1\right)\)\(=\sqrt{\dfrac{1}{2}}\left(\sqrt{7}-1-\sqrt{7}-1\right)=\sqrt{\dfrac{1}{2}}.-2=-\sqrt{2}\)
b)\(\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{7+4\sqrt{3}}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{\left(2+\sqrt{3}\right)^2}}}\)\(=\sqrt{5\sqrt{3}+5\sqrt{48-10\left(2+\sqrt{3}\right)}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{28-10\sqrt{3}}}\)\(=\sqrt{5\sqrt{3}+5\sqrt{\left(5-\sqrt{3}\right)^2}}\)
\(=\sqrt{5\sqrt{3}+5\left(5-\sqrt{3}\right)}\)\(=\sqrt{5\sqrt{3}+25-5\sqrt{3}}=5\)
c) \(\left(4+\sqrt{15}\right)\left(\sqrt{10}-\sqrt{6}\right)\sqrt{4}-\sqrt{15}\)
\(=\left(8+2\sqrt{3}.\sqrt{5}\right).\sqrt{2}\left(\sqrt{5}-\sqrt{3}\right)-\sqrt{15}\)
\(=\left(\sqrt{3}+\sqrt{5}\right)^2\left(\sqrt{5}-\sqrt{3}\right)\sqrt{2}-\sqrt{15}\)
\(=\left(\sqrt{3}+\sqrt{5}\right)\left[\left(\sqrt{5}\right)^2-\left(\sqrt{3}\right)^2\right]\sqrt{2}-\sqrt{15}\)
\(=\left(\sqrt{3}+\sqrt{5}\right)2\sqrt{2}-\sqrt{15}\)\(=2\sqrt{6}+2\sqrt{10}-\sqrt{15}\)
d)\(\sqrt[4]{17+12\sqrt{2}}=\sqrt[4]{9+2.3.2\sqrt{2}+8}=\sqrt[4]{\left(3+2\sqrt{2}\right)^2}=\sqrt{3+2\sqrt{2}}\)
\(=\sqrt{\left(\sqrt{2}+1\right)^2}=\sqrt{2}+1\)
e)\(\sqrt[3]{6\sqrt{3}+10}=\sqrt[3]{\left(\sqrt{3}\right)^3+9+3.\sqrt{3}+1}=\sqrt[3]{\left(\sqrt{3}+1\right)^3}\)\(=\sqrt{3}+1\)
giúp e với ạ e cảm ơnn
