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9 tháng 7 2020
Sủa lại đề:
\(\frac{3+\sqrt{5}}{\sqrt{10}+\sqrt{3+\sqrt{5}}}-\frac{3-\sqrt{5}}{\sqrt{10}+\sqrt{3-\sqrt{5}}}\)
Đặt \(\hept{\begin{cases}\sqrt{3+\sqrt{5}}=a\\\sqrt{3-\sqrt{5}}=b\end{cases}}\)
Khi đó ta có \(a^2+b^2=6\), \(ab=2\), \(a+b=\sqrt{10}\), \(a-b=\sqrt{2}\), \(a^2-b^2=2\sqrt{5}\)
\(=\frac{a^2}{\sqrt{10}+a}-\frac{b^2}{\sqrt{10}+b}\)
\(=\frac{a^2.\left(\sqrt{10}+b\right)-b^2.\left(\sqrt{10}+a\right)}{\left(\sqrt{10}+a\right).\left(\sqrt{10}+b\right)}\)
\(=\frac{\sqrt{10}a^2+a^2b-\sqrt{10}b^2-ab^2}{10+\sqrt{10}a+\sqrt{10}b+ab}\)
\(=\frac{\sqrt{10}.\left(a^2-b^2\right)+ab.\left(a-b\right)}{10+\sqrt{10}.\left(a+b\right)+ab}\)
\(=\frac{\sqrt{10}.2\sqrt{5}+\sqrt{10}.\sqrt{2}}{10+\sqrt{10}.\sqrt{10}+2}\)
\(=\frac{10\sqrt{2}+2\sqrt{2}}{10+10+2}\)
\(=\frac{12\sqrt{2}}{22}\)
\(=\frac{6\sqrt{2}}{11}\)
9 tháng 7 2020
\(\frac{3+\sqrt{5}}{\sqrt{10}+\sqrt{3+\sqrt{5}}}-\frac{3-\sqrt{5}}{\sqrt{10}+\sqrt{3+\sqrt{5}}}
\)
\(=\frac{3+\sqrt{5}-3-\sqrt{5}}{\sqrt{10}+\sqrt{3+\sqrt{5}}}\)
\(=\frac{0}{\sqrt{10}+\sqrt{3+\sqrt{5}}}\)
\(=0\)
CM
2
NC
25 tháng 8 2020
Ta có:
\(\sqrt{27}-\sqrt{5\frac{1}{3}}+4,5\sqrt{2\frac{2}{3}}+2\sqrt{27}\)
\(=3\sqrt{3}-\sqrt{\frac{16}{3}}+4,5\sqrt{\frac{8}{3}}+6\sqrt{3}\)
\(=9\sqrt{3}+\frac{4\sqrt{3}}{3}+3\sqrt{6}\)
\(=\frac{9\sqrt{6}+31\sqrt{3}}{3}\)
25 tháng 8 2020
\(\sqrt{27}-\sqrt{5\frac{1}{3}}+4,5\sqrt{2\frac{2}{3}}+2\sqrt{27}\)
\(=\sqrt{27}-\sqrt{16.\frac{1}{3}}+4,5.\sqrt{4.\frac{1}{3}}+2\sqrt{27}\)
\(=\sqrt{27}-4\sqrt{\frac{1}{3}}+9\sqrt{\frac{1}{3}}+2\sqrt{27}\)
\(=\sqrt{27}-4\sqrt{\frac{1}{3}}+\sqrt{27}+2\sqrt{27}\)
\(=4\sqrt{27}-4\sqrt{\frac{1}{3}}\)
\(=\sqrt{54}-\sqrt{\frac{2}{3}}\)
Y
14 tháng 8 2023
\(b,\sqrt{2}.\sqrt{7+3\sqrt{5}}-\dfrac{4}{\sqrt{5}-1}\\ =\sqrt{14+6\sqrt{5}}-\dfrac{4}{\sqrt{5}-1}\\ =\sqrt{\sqrt{5^2}+2.3\sqrt{5}+3^2}-\dfrac{4}{\sqrt{5}-1}\\ =\sqrt{\left(\sqrt{5}+3\right)^2}-\dfrac{4}{\sqrt{5}-1}\\ =\left|\sqrt{5}+3\right|-\dfrac{4}{\sqrt{5}-1}\\ =\dfrac{\left(\sqrt{5}+3\right)\left(\sqrt{5}-1\right)-4}{\sqrt{5}-1}\\ =\dfrac{2+2\sqrt{5}-4}{\sqrt{5}-1}\\ =\dfrac{-2+2\sqrt{5}}{\sqrt{5}-1}\\ =\dfrac{2\left(-1+\sqrt{5}\right)}{\sqrt{5}-1}\\ =2\)
\(c,\sqrt{27}-6\sqrt{\dfrac{1}{3}}+\dfrac{\sqrt{3}-3}{\sqrt{3}}\\ =3\sqrt{3}-\dfrac{6}{\sqrt{3}}+\dfrac{\sqrt{3}-3}{\sqrt{3}}\)
\(=\dfrac{3\sqrt{3}.\sqrt{3}-6+\sqrt{3}-3}{\sqrt{3}}\\ =\dfrac{9-6+\sqrt{3}-3}{\sqrt{3}}\\ =\dfrac{\sqrt{3}}{\sqrt{3}}\\ =1\)
\(d,\dfrac{9-2\sqrt{3}}{3\sqrt{6}-2\sqrt{2}}\\ =\dfrac{\left(9-2\sqrt{3}\right)\left(3\sqrt{6}+2\sqrt{2}\right)}{\left(3\sqrt{6}-2\sqrt{2}\right)\left(3\sqrt{6}+2\sqrt{2}\right)}\\ =\dfrac{27\sqrt{6}+18\sqrt{2}-18\sqrt{2}-4\sqrt{6}}{\left(3\sqrt{6}\right)^2-\left(2\sqrt{2}\right)^2}\\ =\dfrac{23\sqrt{6}}{54-8}\\ =\dfrac{23\sqrt{6}}{46}\\ =\dfrac{\sqrt{6}}{2}\)
\(\frac{9\sqrt{5}+3\sqrt{27}}{\sqrt{5}+\sqrt{3}}=\frac{\left(9\sqrt{5}+9\sqrt{3}\right)}{\sqrt{5}+\sqrt{3}}=\frac{9\left(\sqrt{5}+\sqrt{3}\right)}{\sqrt{5}+\sqrt{3}}=9\)
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