Question

Bài 1: Thực hiện phép tính:

a) \(\sqrt{\sqrt{28+16\sqrt{3}}}\)

b) \(\sqrt{7+4\sqrt{3}}\)

c) \(\sqrt{\sqrt{56-24\sqrt{5}}}\)

d) \(\sqrt{12-2\sqrt{11}}\)

e) \(\sqrt{9+4\sqrt{2}}\)

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Answer 1

a ) Để ý thấy \(16\sqrt{3}=2.2\sqrt{3}.4=2.\sqrt{12}.4\) , như vậy , ta sẽ tách :

\(28=12+16\) \(\Rightarrow\sqrt{\sqrt{28+16\sqrt{3}}=\sqrt{\sqrt{12+16+16\sqrt{3}}}}=\sqrt{\sqrt{\left(\sqrt{12}+4\right)^2}}=\sqrt{\sqrt{12}+4}\)

\(=\sqrt{3+2.\sqrt{3}+1}=\sqrt{\left(\sqrt{3}+1\right)^2}=\sqrt{3}+1\)

b ) \(4\sqrt{3}=2.2\sqrt{3}\), tách \(7=4+3\)

c ) \(24\sqrt{5}=2.\sqrt{5}.12=2.\sqrt{5}.2.6=2.\sqrt{20}.6\) , tách : \(56=20+36\)

d ) \(2\sqrt{11}=2.11.1\) , tách : \(12=11+1\)

e ) \(4\sqrt{2}=2.\sqrt{2}.2.1=2.\sqrt{8}.1\) , tách : \(9=8+1\)

Answer 2

a) \(\sqrt{\sqrt{28+16\sqrt{3}}}\)

\(=\sqrt{\sqrt{\left(2\sqrt{3}\right)^2+2\cdot2\sqrt{3}\cdot4+16}}\)

\(=\sqrt{\sqrt{\left(2\sqrt{3}+4\right)^2}}\) \(=\sqrt{2\sqrt{3}+4}\)

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

b)\(\sqrt{7+4\sqrt{3}}=\sqrt{4+4\sqrt{3}+3}\)

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

c) \(\sqrt{\sqrt{56-24\sqrt{5}}}=\sqrt{\sqrt{\left(2\sqrt{5}\right)^2-2\cdot2\sqrt{5}\cdot6}+36}\)

\(=\sqrt{\sqrt{\left(2\sqrt{5}-6\right)^2}}=\sqrt{6-2\sqrt{5}}\)

\(=\sqrt{5-2\sqrt{5}+1}=\sqrt{\left(\sqrt{5}-1\right)^2}=\sqrt{5}-1\)

d) \(\sqrt{12-2\sqrt{11}}=\sqrt{11-2\sqrt{11}+1}\)

\(=\sqrt{\left(\sqrt{11}-1\right)^2}=\sqrt{11}-1\)

e) \(\sqrt{9+4\sqrt{2}}=\sqrt{\left(2\sqrt{2}\right)^2+2\cdot2\sqrt{2}+1}\)

\(=\sqrt{\left(2\sqrt{2}+1\right)^2}=2\sqrt{2}+1\)