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Ta có: \(a+b\sqrt{3}=\sqrt{7-4\sqrt{3}}-\sqrt{7+4\sqrt{3}}\)
\(\Leftrightarrow a+b\sqrt{3}=2-\sqrt{3}-2-\sqrt{3}\)
\(\Leftrightarrow a+b\sqrt{3}=-2\sqrt{3}\)
\(\Leftrightarrow a=0;b=-2\)
T=a+b=0+(-2)=-2
\(S=\sqrt{\left(\sqrt{3}\right)^2-2\cdot2\sqrt{3}+2^2}-\sqrt{\left(\sqrt{3}\right)^2+2\cdot2\cdot\sqrt{3}+2^2}\)
\(S=\sqrt{\left(\sqrt{3}-2\right)^2}-\sqrt{\left(\sqrt{3}+2\right)^2}\)
\(S=\left|\sqrt{3}-2\right|-\left|\sqrt{3}+2\right|=-\sqrt{3}+2-\sqrt{3}-2=0+\left(-2\right)\sqrt{3}\)
\(a=0,b=-2\)
\(T=0+-2=-2\)
1: \(=\dfrac{\sqrt{8+2\sqrt{7}}+\sqrt{8-2\sqrt{7}}}{\sqrt{2}}\)
\(=\dfrac{\sqrt{7}+1+\sqrt{7}-1}{\sqrt{2}}=\dfrac{2\sqrt{7}}{\sqrt{2}}=\sqrt{14}\)
3: \(=\sqrt{6+2\sqrt{2\cdot\sqrt{3-\sqrt{3}-1}}}\)
\(=\sqrt{6+2\sqrt{2\cdot\sqrt{2-\sqrt{3}}}}\)
\(=\sqrt{6+2\sqrt{\sqrt{2}\left(\sqrt{3}-1\right)}}\)
\(=\sqrt{6+2\sqrt{\sqrt{6}-\sqrt{2}}}\)
1: \(=\dfrac{\sqrt{8+2\sqrt{7}}+\sqrt{8-2\sqrt{7}}}{\sqrt{2}}\)
\(=\dfrac{\sqrt{7}+1+\sqrt{7}-1}{\sqrt{2}}=\dfrac{2\sqrt{7}}{\sqrt{2}}=\sqrt{14}\)
3: \(=\sqrt{6+2\sqrt{2\cdot\sqrt{3-\sqrt{3}-1}}}\)
\(=\sqrt{6+2\sqrt{2\cdot\sqrt{2-\sqrt{3}}}}\)
\(=\sqrt{6+2\sqrt{\sqrt{2}\left(\sqrt{3}-1\right)}}\)
\(=\sqrt{6+2\sqrt{\sqrt{6}-\sqrt{2}}}\)
a) A=12\(\sqrt{3}\)
B= \(\frac{8}{3}\)
c) C= 1
d)...
Chúc bạn học tốt nha ^^!
a: \(A=\dfrac{\left(\sqrt{7}+\sqrt{3}\right)^2+\left(\sqrt{7}-\sqrt{3}\right)^2}{4}\)
\(=\dfrac{10+2\sqrt{21}+10-2\sqrt{21}}{4}=\dfrac{20}{4}=5\)
b: \(B=6\sqrt{3}+\sqrt{3}-1-2\sqrt{2}\)
\(=7\sqrt{3}-2\sqrt{2}-1\)
\(a,\dfrac{3}{\sqrt{7}-4}+\dfrac{4+\sqrt{7}}{3}\)
\(=\dfrac{9}{3\left(\sqrt{7}-4\right)}+\dfrac{\left(\sqrt{7}-4\right)\left(\sqrt{7}+4\right)}{3\left(\sqrt{7}-4\right)}\)
\(=\dfrac{9+7-16}{3\left(\sqrt{7}-4\right)}\)
\(=0\)
\(b,\left(\dfrac{\sqrt{6}-\sqrt{2}}{\sqrt{3}-1}+\dfrac{1}{\sqrt{3}+\sqrt{2}}\right):\dfrac{1}{2\sqrt{3}}\)
\(=\left[\dfrac{\sqrt{2}\left(\sqrt{3}-1\right)}{\sqrt{3}-1}+\dfrac{\sqrt{3}-\sqrt{2}}{\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}+\sqrt{2}\right)}\right]\cdot2\sqrt{3}\)
\(=\left(\sqrt{2}+\dfrac{\sqrt{3}-\sqrt{2}}{3-2}\right)\cdot2\sqrt{3}\)
\(=\left(\sqrt{2}+\sqrt{3}-\sqrt{2}\right)\cdot2\sqrt{3}\)
\(=\sqrt{3}\cdot2\sqrt{3}\)
\(=6\)
#\(Toru\)
ta có : \(A=\dfrac{3+\sqrt{5}}{2\sqrt{2}+\sqrt{3+\sqrt{5}}}+\dfrac{3-\sqrt{5}}{2\sqrt{2}-\sqrt{3-\sqrt{5}}}\)
\(=\dfrac{\sqrt{10}+3\sqrt{2}}{5+\sqrt{5}}+\dfrac{\sqrt{10}-3\sqrt{2}}{5-\sqrt{5}}\) \(=\dfrac{4\sqrt{2}}{\sqrt{5}\left(\sqrt{5}+1\right)\left(\sqrt{5}-1\right)}=\dfrac{4\sqrt{2}}{4\sqrt{5}}=\sqrt{\dfrac{2}{5}}\)
làm tương tự với B rồi --> ...
a) \(\sqrt{2}\left(\sqrt{4+\sqrt{7}}+\sqrt{4-\sqrt{7}}\right)\)
\(=\sqrt{2\cdot\left(4+\sqrt{7}\right)}+\sqrt{2\cdot\left(4-\sqrt{7}\right)}\)
\(=\sqrt{8+2\sqrt{7}}+\sqrt{8-2\sqrt{7}}\)
\(=\sqrt{\left(\sqrt{7}\right)^2+2\cdot\sqrt{7}\cdot1+1^2}+\sqrt{\left(\sqrt{7}\right)^2-2\cdot\sqrt{7}\cdot1+1^2}\)
\(=\sqrt{\left(\sqrt{7}+1\right)^2}+\sqrt{\left(\sqrt{7}-1\right)^2}\)
\(=\left|\sqrt{7}+1\right|+\left|\sqrt{7}-1\right|\)
\(=\sqrt{7}+1+\sqrt{7}-1\)
\(=2\sqrt{7}\)
b) \(\sqrt{2-\sqrt{3}}-\sqrt{2+\sqrt{3}}\)
\(=\dfrac{\sqrt{2}\cdot\left(\sqrt{2-\sqrt{3}}-\sqrt{2+\sqrt{3}}\right)}{\sqrt{2}}\)
\(=\dfrac{\sqrt{2\cdot\left(2-\sqrt{3}\right)}-\sqrt{2\cdot\left(2+\sqrt{3}\right)}}{\sqrt{2}}\)
\(=\dfrac{\sqrt{4-2\sqrt{3}}-\sqrt{4+2\sqrt{3}}}{\sqrt{2}}\)
\(=\dfrac{\sqrt{\left(\sqrt{3}\right)^2-2\cdot\sqrt{3}\cdot1+1^2}-\sqrt{\left(\sqrt{3}\right)^2+2\cdot\sqrt{3}\cdot1+1^2}}{\sqrt{2}}\)
\(=\dfrac{\sqrt{\left(\sqrt{3}-1\right)^2}-\sqrt{\left(\sqrt{3}+1\right)^2}}{\sqrt{2}}\)
\(=\dfrac{\left|\sqrt{3}-1\right|-\left|\sqrt{3}+1\right|}{\sqrt{2}}\)
\(=\dfrac{\sqrt{3}-1-\sqrt{3}-1}{ }\)
\(=-\dfrac{2}{\sqrt{2}}\)
\(=-\sqrt{2}\)
\(S^3=\left(\sqrt[3]{7+4\sqrt{3}+}\sqrt[3]{7-4\sqrt{3}}\right)^3\)
= \(7+4\sqrt{3}+7-4\sqrt{3}+3.\sqrt{7+4\sqrt{3}}.\sqrt{7-4\sqrt{3}}.\left(a+b\right)\)
= 14+\(3.\sqrt{49-48}.S\)
= 14+3S
=> S3-3S=14+3S-3S=14
\(P=S^3-3S\)
\(P=\left(\sqrt[3]{7+4\sqrt{3}}+\sqrt[3]{7-4\sqrt{3}}\right)^3-3\left(\sqrt[3]{7+4\sqrt{3}}+\sqrt[3]{7-4\sqrt{3}}\right)\)
\(P=7+4\sqrt{3}+3\left(\sqrt[3]{7+4\sqrt{3}}\right)^2.\sqrt[3]{7-4\sqrt{3}}+3.\sqrt[3]{7+4\sqrt{3}}\left(\sqrt[3]{7-4\sqrt{3}}\right)^2+7-4\sqrt{3}\text{}\text{}-3\left(\sqrt[3]{7+4\sqrt{3}}+\sqrt[3]{7-4\sqrt{3}}\right)\)
\(P=14+3\sqrt[3]{7+4\sqrt{3}}.\sqrt[3]{7-4\sqrt{3}}\text{}\left(\sqrt[3]{7+4\sqrt{3}}+\sqrt[3]{7-4\sqrt{3}}\right)\text{}\text{}-3\left(\sqrt[3]{7+4\sqrt{3}}+\sqrt[3]{7-4\sqrt{3}}\right)\)
\(P=14+3\sqrt[3]{\left(7+4\sqrt{3}\right)\left(7-4\sqrt{3}\right)}\text{}\left(\sqrt[3]{7+4\sqrt{3}}+\sqrt[3]{7-4\sqrt{3}}\right)\text{}\text{}-3\left(\sqrt[3]{7+4\sqrt{3}}+\sqrt[3]{7-4\sqrt{3}}\right)\)
\(P=14+3\sqrt[3]{49-48}\text{}\left(\sqrt[3]{7+4\sqrt{3}}+\sqrt[3]{7-4\sqrt{3}}\right)\text{}\text{}-3\left(\sqrt[3]{7+4\sqrt{3}}+\sqrt[3]{7-4\sqrt{3}}\right)\)
\(P=14+3\text{}\left(\sqrt[3]{7+4\sqrt{3}}+\sqrt[3]{7-4\sqrt{3}}\right)\text{}\text{}-3\left(\sqrt[3]{7+4\sqrt{3}}+\sqrt[3]{7-4\sqrt{3}}\right)\)
\(P=14\)