\(10+\sqrt{60}+\sqrt{24}+\sqrt{40}=10+2\sqrt{15}+2\sqrt{6}+2\sqrt{10}\)

\(=\left(5+2\sqrt{15}+3\right)+2+2\sqrt{2}\left(\sqrt{5}+\sqrt{3}\right)\)

\(=\left(\sqrt{5}+\sqrt{3}\right)^2+2\sqrt{2}\left(\sqrt{5}+\sqrt{3}\right)+2\)

\(=\left(\sqrt{5}+\sqrt{3}+\sqrt{2}\right)^2\)

\(\Rightarrow\sqrt{10+\sqrt{60}+\sqrt{24}+\sqrt{40}}=\sqrt{5}+\sqrt{3}+\sqrt{2}\)

Dùng hẳng đẳng thức 3 số:

$(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca$
$VT=\sqrt{5+3+2+2\sqrt{15}+2\sqrt{6}+2\sqrt{10}}=\sqrt{(\sqrt5+\sqrt3+\sqrt2)^2}=VP(đpcm)$

\(\sqrt{10+\sqrt{24}+\sqrt{40}+\sqrt{60}}=\sqrt{2+3+5+2\sqrt{2.3}+2\sqrt{2.5}+2\sqrt{3.5}}\)

\(=\sqrt{\left(\sqrt{2}+\sqrt{3}+\sqrt{5}\right)^2}=\sqrt{2}+\sqrt{3}+\sqrt{5}\)

22,13177916

\(\sqrt{10+\sqrt{24}+\sqrt{40}+\sqrt{60}}\)

\(=\sqrt{10+\sqrt{25-1}+\sqrt{49-9}+\sqrt{64-4}}\)

\(=\sqrt{10+\sqrt{5^2-1^2}+\sqrt{7^2-3^2}+\sqrt{8^2-2^2}}\)

\(=\sqrt{10+\left(\sqrt{5-1}\right)^2+\left(\sqrt{7-3}\right)^2+\left(\sqrt{8-2}\right)^2}\)

\(=\sqrt{10+5-1+7-3+8-2}\)

\(=\sqrt{24}\)

\(=2\sqrt{6}\)

Muội chả hỉu sao tỷ học giỏi vậy!

\(\sqrt{10+\sqrt{60}-\sqrt{24}-\sqrt{40}}\)

\(=\sqrt{2+3+5+2\sqrt{6}+2\sqrt{10}+2\sqrt{15}}\)

\(=\left(\sqrt{3}+\sqrt{5}-\sqrt{2}\right)^2\)

\(=\sqrt{3}+\sqrt{5}-\sqrt{2}\)

\(\sqrt{10+2\sqrt{15}+2\sqrt{6}+2\sqrt{10}}\)

\(=\sqrt{5+3+2+2\sqrt{5}.\sqrt{3}+2\sqrt{3}.\sqrt{2}+2\sqrt{5}.\sqrt{2}}\)

\(=\sqrt{\left(\sqrt{5}+\sqrt{3}+\sqrt{2}\right)^2}=\sqrt{5}+\sqrt{3}+\sqrt{2}\)