Question

Rút gọn
a)\(\sqrt{13-4\sqrt{3}}\)
b)\(\sqrt{9+6\sqrt{2}}\)
c)\(\sqrt{10+4\sqrt{6}}\)
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Answer 1

a)

\(\sqrt{13-4\sqrt{3}}\\ =\sqrt{12-2\cdot2\sqrt{3}\cdot1+1}\\ =\sqrt{\left(2\sqrt{3}\right)^2-2\cdot2\sqrt{3}\cdot1+1}\\ =\sqrt{\left(2\sqrt{3}-1\right)^2}\\ =2\sqrt{3}-1\)

b)

\(\sqrt{9+6\sqrt{2}}\\ =\sqrt{9+2\cdot\sqrt{3}\cdot\sqrt{3}\cdot\sqrt{2}}\\ =\sqrt{6+2\cdot\sqrt{6}\cdot\sqrt{3}+3}\\ =\sqrt{\left(\sqrt{6}\right)^2+2\cdot\sqrt{6}\cdot\sqrt{3}+\left(\sqrt{3}\right)^2}\\ =\sqrt{\left(\sqrt{6}+\sqrt{3}\right)^2}\\ =\sqrt{6}+\sqrt{3}=\sqrt{3}\left(\sqrt{2}+1\right)\)

c)

\(\sqrt{10+4\sqrt{6}}\\ =\sqrt{6+2\cdot\sqrt{6}\cdot2+4}\\ =\sqrt{\left(\sqrt{6}\right)^2+2\cdot\sqrt{6}\cdot2+2^2}\\ =\sqrt{\left(\sqrt{6}+2\right)^2}\\ =\sqrt{6}+2\)