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Bài làm:
\(\sqrt{14-8\sqrt{3}}-\sqrt{24-12\sqrt{3}}\)
\(=\sqrt{8-8\sqrt{3}+6}-\sqrt{18-12\sqrt{3}+6}\)
\(=\sqrt{\left(2\sqrt{2}-\sqrt{6}\right)^2}-\sqrt{\left(3\sqrt{2}-\sqrt{6}\right)^2}\)
\(=2\sqrt{2}-\sqrt{6}-3\sqrt{2}+\sqrt{6}\)
\(=-\sqrt{2}\)
Học tốt
a,\(\sqrt{\left(\sqrt{3}-1\right)^2}\) \(+\sqrt{\left(\sqrt{3}+1\right)^2}=2\sqrt{3}\)
b. \(\sqrt{\left(2\sqrt{5}+2\right)^2}+\sqrt{\left(\sqrt{5}-2\right)^2}=3\sqrt{5}\)
c,\(\sqrt{\left(3-2\sqrt{2}\right)^2}+\sqrt{\left(2\sqrt{2}+1\right)^2}=4\)
d.\(\sqrt{\left(2-\sqrt{2}\right)^2}+\sqrt{\left(3\sqrt{2}-2\right)^2}=2\sqrt{2}\)
\(\sqrt{14-8\sqrt{3}}-\sqrt{24-12\sqrt{3}}=\dfrac{\sqrt{16-2.4.2\sqrt{3}+12}-\sqrt{36-2.6.2\sqrt{3}+12}}{\sqrt{2}}=\dfrac{\sqrt{\left(4-2\sqrt{3}\right)^2}-\sqrt{\left(6-2\sqrt{3}\right)^2}}{\sqrt{2}}=\dfrac{4-2\sqrt{3}-6+2\sqrt{3}}{\sqrt{2}}=\dfrac{-2}{\sqrt{2}}=-\sqrt{2}\)
\(\frac{9\sqrt{6}-12\sqrt{3}}{3\sqrt{6}-3\sqrt{3}}=\frac{3\sqrt{3}\left(3\sqrt{2}-4\right)}{\sqrt{3}\left(\sqrt{2}-1\right)}=\frac{3\sqrt{2}-4}{\sqrt{2}-1}\)
\(a,\frac{13\sqrt{12}-4\sqrt{6}}{24-4\sqrt{3}}=\frac{13\sqrt{12}-\sqrt{8}.\sqrt{12}}{2\sqrt{12}.\sqrt{12}}=\frac{13-\sqrt{8}}{2\sqrt{12}}\)
\(\sqrt{3-\sqrt{5}}+\sqrt{3+\sqrt{5}}\)
\(=\sqrt{\frac{6-2\sqrt{5}}{2}}+\sqrt{\frac{6+2\sqrt{5}}{2}}\)
\(=\sqrt{\frac{5-2\sqrt{5}+1}{2}}+\sqrt{\frac{5+2\sqrt{5}+1}{2}}\)
\(=\frac{\sqrt{\left(\sqrt{5}-1\right)^2}}{\sqrt{2}}+\frac{\sqrt{\left(\sqrt{5}+1\right)^2}}{\sqrt{2}}\)
\(=\frac{\sqrt{5}-1}{\sqrt{2}}+\frac{\sqrt{5}+1}{\sqrt{2}}\)
\(=\frac{\sqrt{5}-1+\sqrt{5}+1}{\sqrt{2}}=\frac{2\sqrt{5}}{\sqrt{2}}=\frac{\sqrt{2}.\sqrt{2}.\sqrt{5}}{\sqrt{2}}=\sqrt{10}\)
\(\sqrt{24+8\sqrt{5}}+\) \(\sqrt{9-4\sqrt{5}}=\) \(\sqrt{\left(2\sqrt{5}\right)^2+2.2\sqrt{5}.2+4}\) + \(\sqrt{5-2\sqrt{5}.2+4}\)
= \(\sqrt{\left(2\sqrt{5}+2\right)^2}+\) \(\sqrt{\left(\sqrt{5}-2\right)^2}\) = \(2\sqrt{5}+2+\sqrt{5}-2=3\sqrt{5}\)
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\(\sqrt{\sqrt{5}-\sqrt{3-\sqrt{29-12\sqrt{5}}}}\) = \(\sqrt{\sqrt{5}-\sqrt{3-\left(2\sqrt{5}-3\right)}}\)= \(\sqrt{\sqrt{5}-\sqrt{6-2\sqrt{5}}}=\sqrt{\sqrt{5}-\sqrt{5}+1}=1\)
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\(\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}=\sqrt{13+30\sqrt{2+2\sqrt{2}+1}}\)
= \(\sqrt{13+30\sqrt{3+2\sqrt{2}}}=\sqrt{13+30\left(\sqrt{2}+1\right)}=\sqrt{43+30\sqrt{2}}\) \(=\sqrt{\left(3\sqrt{2}+5\right)^2}=3\sqrt{2}+5\)
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Lời giải:
\(\sqrt{14-8\sqrt{3}}-\sqrt{24-12\sqrt{3}}=\sqrt{8+6-2\sqrt{8.6}}-\sqrt{18+6-2\sqrt{18.6}}\)
\(=\sqrt{(\sqrt{8}-\sqrt{6})^2}-\sqrt{(\sqrt{18}-\sqrt{6})^2}\)
\(=\sqrt{8}-\sqrt{6}-(\sqrt{18}-\sqrt{6})=\sqrt{8}-\sqrt{6}\)