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a) \(M=\sqrt{3-2\sqrt{2}}+\sqrt{6+4\sqrt{2}}=\sqrt{2-2\sqrt{2}+1}+\sqrt{2+2.\sqrt{2}.2+4}=\sqrt{\left(\sqrt{2}-1\right)^2}+\sqrt{\left(\sqrt{2}+2\right)^2}=\left|\sqrt{2}-1\right|+\sqrt{2}+2=\sqrt{2}-1+\sqrt{2}+2=2\sqrt{2}+1\)
b) \(N=\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}}=\dfrac{\sqrt{4+2\sqrt{3}}+\sqrt{4-2\sqrt{3}}}{\sqrt{2}}=\dfrac{\sqrt{3+2\sqrt{3}+1}+\sqrt{3-2\sqrt{3}+1}}{\sqrt{2}}=\dfrac{\sqrt{\left(\sqrt{3}+1\right)^2}+\sqrt{\left(\sqrt{3}-1\right)^2}}{\sqrt{2}}=\dfrac{\sqrt{3}+1+\left|\sqrt{3}-1\right|}{\sqrt{2}}=\dfrac{\sqrt{3}+1+\sqrt{3}-1}{\sqrt{2}}=\dfrac{2\sqrt{3}}{\sqrt{2}}=\sqrt{2}.\sqrt{3}=\sqrt{6}\)
Hướng dẫn trả lời:
M=√3−2√2−√6+4√2=√(√2)2−2√2.1+12−√(2)2+2√2+(√2)2=√(√2−1)3−√(2+√2)2=∣∣√2−1∣∣−∣∣2+√2∣∣=√2−1−2−√2=−3M=3−22−6+42=(2)2−22.1+12−(2)2+22+(2)2=(2−1)3−(2+2)2=|2−1|−|2+2|=2−1−2−2=−3
N=√2+√3+√2−√3⇒N2=(√2+√3+√2−√3)2=2+√3+2√(2+√3)(2−√3)+2−√3=4+2√4−3=6N=2+3+2−3⇒N2=(2+3+2−3)2=2+3+2(2+3)(2−3)+2−3=4+24−3=6
Vì N > 0 nên N2 = 6 ⇒ N = √6. Vậy
\(b,\frac{2+\sqrt{3}}{1-\sqrt{4-2\sqrt{3}}}+\frac{2-\sqrt{3}}{1+\sqrt{4+2\sqrt{3}}}\)
\(=\frac{2+\sqrt{3}}{1-\sqrt{3-2\sqrt{3}+1}}+\frac{2-\sqrt{3}}{1+\sqrt{3+2\sqrt{3}+1}}\)
\(=\frac{2+\sqrt{3}}{1-\sqrt{\left(\sqrt{3}-1\right)^2}}+\frac{2-\sqrt{3}}{1+\sqrt{\left(\sqrt{3}+1\right)^2}}\)
\(=\frac{2+\sqrt{3}}{1-\left(\sqrt{3}-1\right)}+\frac{2-\sqrt{3}}{1+\sqrt{3}+1}\)
\(=\frac{2+\sqrt{3}}{2-\sqrt{3}}+\frac{2-\sqrt{3}}{2+\sqrt{3}}\)
\(=\frac{\left(2+\sqrt{3}\right)^2}{\left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right)}+\frac{\left(2-\sqrt{3}\right)^2}{\left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right)}\)
\(=\frac{4+4\sqrt{3}+3+4-4\sqrt{3}+3}{4-3}\)
\(=14\)
\(a,\frac{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+\sqrt{16}}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)
\(=\frac{\sqrt{2}+\sqrt{3}+4+\sqrt{6}+\sqrt{8}}{\sqrt{2}+\sqrt{3}+2}\)
\(=\frac{\sqrt{2}+\sqrt{3}+2}{\sqrt{2}+\sqrt{3}+2}+\frac{\sqrt{2}.\sqrt{2}+\sqrt{2}.\sqrt{3}+\sqrt{2}.2}{\sqrt{2}+\sqrt{3}+2}\)
\(=1+\frac{\sqrt{2}\left(\sqrt{2}+\sqrt{3}+2\right)}{\sqrt{2}+\sqrt{3}+2}\)
\(=1+\sqrt{2}\)
mình nghĩ bài này sai đề,
ĐÚng phải là\(\sqrt[3]{2+\sqrt{3}}\)
( KHÔNG CHẮC NỮA :D )
a) \(\frac{\sqrt{6}+\sqrt{14}}{2\sqrt{3}+\sqrt{28}}=\frac{\sqrt{2}\left(\sqrt{3}+\sqrt{7}\right)}{2\left(\sqrt{3}+\sqrt{7}\right)}=\frac{1}{\sqrt{2}}\)
b) \(\frac{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+\sqrt{16}}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)
\(=\frac{\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)+\left(\sqrt{4}+\sqrt{6}+\sqrt{8}\right)}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)
\(=\frac{\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)\left(1+\sqrt{2}\right)}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)
\(=1+\sqrt{2}\)
\(\frac{4+\sqrt{2}-\sqrt{3}-\sqrt{6}+\sqrt{8}}{2+\sqrt{2}-\sqrt{3}}\)
\(=\frac{\left(\sqrt{4}+\sqrt{2}\right)-\left(\sqrt{3}+\sqrt{6}\right)+\left(\sqrt{4}+\sqrt{8}\right)}{2+\sqrt{2}-\sqrt{3}}\) ( Tách 4 thành \(\sqrt{4}+\sqrt{4}\) )
\(=\frac{\sqrt{2}\left(\sqrt{2}+1\right)-\sqrt{3}\left(1+\sqrt{2}\right)+\sqrt{4}\left(1+\sqrt{2}\right)}{2+\sqrt{2}-\sqrt{3}}\)
\(=\frac{\left(\sqrt{2}-\sqrt{3}+2\right)\left(\sqrt{2}+1\right)}{2+\sqrt{2}-\sqrt{3}}\)
\(=\sqrt{2}+1\)
\(N=\sqrt{6+2\sqrt{2}.\sqrt{3+\sqrt{4-2\sqrt{3}}}}=\sqrt{6+2\sqrt{2}.\sqrt{3+\sqrt{3-2\sqrt{3}+1}}}\)
\(=\sqrt{6+2\sqrt{2}.\sqrt{3+\sqrt{\left(\sqrt{3}-1\right)^2}}}\)\(=\sqrt{6+2\sqrt{2}.\sqrt{3+\sqrt{3}-1}=}\)
\(=\sqrt{6+2\sqrt{4-2\sqrt{3}}}=\sqrt{6+2\left(\sqrt{3}-1\right)}\) \(=\sqrt{4+2\sqrt{3}}=\sqrt{3}+1\)