CMR \(\frac{\sqrt[4]{5}+1}{\sqrt[4]{5}-1}=\sqrt[4]{\frac{3+2\sqrt[4]{5}}{3-2\sqrt[4]{5}}}\)
Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\frac{5\sqrt{7}-7\sqrt{5}+2\sqrt{70}}{\sqrt{35}}\)
\(=\frac{\sqrt{35}.(5\sqrt{7}-7\sqrt{5}+2\sqrt{70})}{\sqrt{35}.\sqrt{35}}\)
\(=\frac{\sqrt{35}.(5\sqrt{7}-7\sqrt{5}+2\sqrt{70})}{35}\)
\(\sqrt{\frac{4}{3}}+\sqrt{12}-\frac{4}{3}\sqrt{\frac{3}{4}}\)
\(=\frac{\sqrt{4}}{\sqrt{3}}+\sqrt{12}-\frac{4}{3}\cdot\frac{\sqrt{3}}{\sqrt{4}}\)
\(=\frac{2\sqrt{3}}{\sqrt{3}.\sqrt{3}}+\sqrt{12}-\frac{4}{3}\cdot\frac{\sqrt{3}}{2}\)
\(=\frac{2\sqrt{3}}{3}+2\sqrt{3}-\frac{2\sqrt{3}}{3}\)
\(=2\sqrt{3}\left(\frac{1}{3}+1-\frac{1}{3}\right)\)
\(=2\sqrt{3}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(C=\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{7+4\sqrt{3}}}}}\)
\(C=\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{7+2\sqrt{12}}}}}\)
\(C=\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{48-10\left(2+\sqrt{3}\right)}}}\)
\(C=\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{28-10\sqrt{3}}}}\)
\(C=\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{28-2\sqrt{75}}}}\)
\(C=\sqrt{4+\sqrt{5\sqrt{3}+5\left(5-\sqrt{3}\right)}}\)
\(C=\sqrt{4+5}\)
\(C=3\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\sqrt[4]{\frac{3-2\sqrt[4]{5}}{3+2\sqrt[4]{5}}}\)= \(\sqrt{\frac{\sqrt{5}-2}{3+2\sqrt[4]{5}}}\)
Từ đó thì
\(\frac{\sqrt[4]{5}-1}{\sqrt[4]{5}+1}\)= \(\sqrt{\frac{\sqrt{5}-2}{3+2\sqrt[4]{5}}}\)
<=> \(\frac{1+\sqrt{5}-2\sqrt[4]{5}}{1+\sqrt{5}+2\sqrt[4]{5}}=\frac{\sqrt{5}-2}{3-2\sqrt[4]{5}}\)
<=> \(3-\sqrt{5}-4\sqrt[4]{5}+2\sqrt{5}\sqrt[4]{5}\) = \(3-\sqrt{5}-4\sqrt[4]{5}+2\sqrt{5}\sqrt[4]{5}\)
Vậy cái đầu tiên là đúng
Đặt \(a=\sqrt[4]{5}\Leftrightarrow5=a^4\)
Ta cần chứng minh: \(\left(\frac{a+1}{a-1}\right)^4=\frac{3+2a}{3-2a}\)
Khai triển: \(VT=\left(\frac{a+1}{a-1}\right)^4=\frac{\left(a+1\right)^4}{\left(a-1\right)^4}\)
\(=\frac{2\left(3+2a\right).\left(1+a^2\right)}{2\left(3-2a\right).\left(1+a^2\right)}\)
\(\frac{3+2a}{3-2a}=VP\)(đpcm)