\(\frac{5+\sqrt{5}}{5-\sqrt{5}}\)+\(\frac{5-\sqrt{5}}{5+\sqrt{5}}\)Thu gọn
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Thu gọn biểu thức
\(y=\frac{\sqrt{4+\sqrt{5}}+\sqrt{4-\sqrt{5}}}{\sqrt{4+\sqrt{11}}}-\frac{\sqrt{20-4\sqrt{23}}}{\sqrt{5+\sqrt{2}}-\sqrt{5-\sqrt{2}}}+\frac{8}{3+\sqrt{5}}\)
Thu gọn \(P=\frac{3+\sqrt{5}}{\sqrt{10}+\sqrt{3+\sqrt{5}}}\)- \(\frac{3-\sqrt{5}}{\sqrt{10}+\sqrt{3-\sqrt{5}}}\)
HELPPPPP... PLZZZZZZZZ <3
\(P=\frac{3\sqrt{2}+\sqrt{10}}{2\sqrt{5}+\sqrt{6+2\sqrt{5}}}-\frac{3\sqrt{2}-\sqrt{10}}{2\sqrt{5}+\sqrt{6-2\sqrt{5}}}=\frac{3\sqrt{2}+\sqrt{10}}{2\sqrt{5}+\sqrt{5}+1}-\frac{3\sqrt{2}-\sqrt{10}}{2\sqrt{5}+\sqrt{5}-1}\)
\(=\frac{3\sqrt{2}+\sqrt{10}}{3\sqrt{5}+1}-\frac{3\sqrt{2}-\sqrt{10}}{3\sqrt{5}-1}=\frac{9\sqrt{10}-3\sqrt{2}+15\sqrt{2}-\sqrt{10}-9\sqrt{10}-3\sqrt{2}+15\sqrt{2}+\sqrt{10}}{44}\)
\(=\frac{24\sqrt{2}}{44}=\frac{6\sqrt{2}}{11}.\)
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Thu gọn biểu thức sau:
\(B=5\left(\sqrt{2+\sqrt{3}}+\sqrt{3-\sqrt{5}}-\sqrt{\frac{5}{2}}\right)^2-\left(\sqrt{2-\sqrt{3}}+\sqrt{3+\sqrt{5}}-\sqrt{\frac{3}{2}}\right)^2\)
rút gọn
A= \(\frac{\sqrt{3}+\sqrt{5}}{\sqrt{3}-\sqrt{5}}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}\)
B= \(\frac{5+2\sqrt{5}}{\sqrt{5}}+\frac{3+\sqrt{3}}{\sqrt{3}}-\left(\sqrt{5}+\sqrt{3}\right)\)
\(A=\frac{\left(\sqrt{3}+\sqrt{5}\right)\left(\sqrt{3}+\sqrt{5}\right)}{\left(\sqrt{3}-\sqrt{5}\right)\left(\sqrt{3}+\sqrt{5}\right)}+\frac{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}\\ A=\frac{3+2\sqrt{15}+5}{3-5}+\frac{5-2\sqrt{15}+3}{5-3}\\ A=\frac{3+2\sqrt{15}+5-\left(5-2\sqrt{15}+3\right)}{-2}\\ A=\frac{4\sqrt{15}}{-2}=-2\sqrt{15}\)
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\(B=\frac{\sqrt{5}\left(5+2\sqrt{5}\right)}{\left(\sqrt{5}\right)^2}+\frac{\sqrt{3}\left(3+\sqrt{3}\right)}{\left(\sqrt{3}\right)^2}-\left(\sqrt{5}+\sqrt{3}\right)\\ B=\frac{5\left(\sqrt{5}+2\right)}{5}+\frac{3\left(\sqrt{3}+1\right)}{3}-\left(\sqrt{5}+\sqrt{3}\right)\\ B=\sqrt{5}+2+\sqrt{3}+1-\sqrt{5}-\sqrt{3}\\ B=3\)
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Tính (thu gọn) \(\frac{\sqrt{5+2\sqrt{5}}}{\sqrt{6+3\sqrt{5}}}\)
rút gọn biểu thức
a) \(\frac{5+\sqrt{5}}{5-\sqrt{5}}+\frac{5-\sqrt{5}}{5+\sqrt{5}}\)
b) \(\frac{3}{2+\sqrt{3}}+\frac{13}{4-\sqrt{3}}+\frac{6}{\sqrt{3}}\)
c) \(\left(\frac{\sqrt{14}-\sqrt{7}}{\sqrt{2}-1}+\frac{\sqrt{15}-\sqrt{5}}{\sqrt{3}-1}\right):\frac{1}{\sqrt{7}-\sqrt{5}}\)
rút gọn \(\frac{1}{2-\sqrt{5}}+\frac{2}{\sqrt{5}+\sqrt{3}}:\frac{1}{\sqrt{21-12\sqrt{3}}}\)
\(\frac{5\sqrt{3}}{\sqrt{3}-\sqrt{5}-\sqrt{3}}-\frac{5\sqrt{3}}{\sqrt{3-\sqrt{5}}+\sqrt{3}}\)
GIÚP MK NHANH NHẾ
MK TICK CHO
Rút gọn
\(A=\frac{1}{\sqrt{3}+1}+\frac{1}{\sqrt{3}-1}\)
\(B=\frac{5+\sqrt{5}}{5-\sqrt{5}}+\frac{5-\sqrt{5}}{5+\sqrt{5}}\)
\(C=\frac{\sqrt{3}}{\sqrt{\sqrt{3}+1}-1}-\frac{\sqrt{3}}{\sqrt{\sqrt{3}+1}+1}\)
\(A=\frac{1}{\sqrt{3}+1}+\frac{1}{\sqrt{3}-1}\)
\(=\frac{\sqrt{3}-1}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}+\frac{\sqrt{3}+1}{\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)}\)
\(=\frac{\sqrt{3}-1+\sqrt{3}+1}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}\)
\(=\frac{2\sqrt{3}}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}\)
\(=\frac{2\sqrt{3}}{3-1}\)
\(=\frac{2\sqrt{3}}{2}\)
\(=\sqrt{3}\)
\(B=\frac{5+\sqrt{5}}{5-\sqrt{5}}+\frac{5-\sqrt{5}}{5+\sqrt{5}}\)
\(=\frac{\sqrt{5}\left(\sqrt{5}+1\right)}{\sqrt{5}\left(\sqrt{5}-1\right)}+\frac{\sqrt{5}\left(\sqrt{5}-1\right)}{\sqrt{5}\left(\sqrt{5}+1\right)}\)
\(=\frac{\left(\sqrt{5}+1\right)}{\left(\sqrt{5}-1\right)}+\frac{\left(\sqrt{5}-1\right)}{\left(\sqrt{5}+1\right)}\)
\(=\frac{\left(\sqrt{5}+1\right)^2}{\left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right)}+\frac{\left(\sqrt{5}-1\right)^2}{\left(\sqrt{5}+1\right)\left(\sqrt{5}-1\right)}\)
\(=\frac{5+2\sqrt{5}+1+5-2\sqrt{5}+1}{\left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right)}\)
\(=\frac{12}{\left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right)}\)
\(=\frac{12}{5-1}\)
\(=\frac{12}{4}\)
\(=3\)
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1) Chứng minh: \(2\sqrt{n}-3< \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{n}}< 2\sqrt{n}-2\forall n\ge2\)
2) Thu gọn: \(A=5\left(\sqrt{2+\sqrt{3}}+\sqrt{3-\sqrt{5}}-\sqrt{\frac{5}{2}}\right)^2+\left(\sqrt{2-\sqrt{3}}+\sqrt{3+\sqrt{5}}-\sqrt{\frac{3}{2}}\right)^2\)