gải phương trình \(\sqrt[3]{x}-3\sqrt[3]{x}=20\)

gải phương trình\(x\sqrt[]{\frac{1}{x}}-2x\sqrt[3]{x}=20\)

Có nhầm đề không vậy? Ở tử có n dấu căn, ở mẫu có n-1

dấu căn . giả sử có một biểu thức bất kì: \(\frac{\sqrt{2+\sqrt{2}}}{\sqrt{2}}>1\)

vậy sao chứng minh?

Đặt \(a=\sqrt{2+\sqrt{2+...+\sqrt{2}}}\)(có n dấu căn )

\(\Rightarrow a^2=2+\sqrt{2+\sqrt{2+...+\sqrt{2}}}\)(có n-1 dấu căn)

\(\Rightarrow\sqrt{2+\sqrt{2+...+\sqrt{2}}}=a^2-2\)(có n-1 dấu căn)

Ta có \(A=\frac{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}\)(ở tử có n dấu căn : ở mẩu có n-1 dấu căn )

\(A=\frac{2-a}{2-\left(a^2-2\right)}=\frac{2-a}{4-a^2}=\frac{1}{a+2}\)

Dễ thấy \(\sqrt{2}a< \sqrt{2+\sqrt{2+...+\sqrt{2+2}}}\)(có n dấu căn)

            \(1,4< a< 2\)

Suy ra \(3,4< a+2< 4\)

\(\frac{1}{3,4}>\frac{1}{a+2}>\frac{1}{4}\)

\(\frac{3}{10}>\frac{1}{a+2}>\frac{1}{4}\)hay\(\frac{1}{4}< A< \frac{3}{10}\)(1)

Từ (1) suy ra ĐPCM

a/ \(D\sqrt{2}=\sqrt{4-2\sqrt{3}}+\sqrt{4+2\sqrt{3}}=\sqrt{\left(\sqrt{3}-1\right)^2}+\sqrt{\left(\sqrt{3}+1\right)^2}\)

\(=\sqrt{3}-1+\sqrt{3}+1=2\sqrt{3}\Rightarrow D=\frac{2\sqrt{3}}{\sqrt{2}}=\sqrt{6}\)

b/\(2E=\sqrt[3]{8\sqrt{5}-16}+\sqrt[3]{8\sqrt{5}+16}\)

\(=\sqrt[3]{5\sqrt{5}-3.5.1+3\sqrt{5}-1}+\sqrt[3]{5\sqrt{5}+3.5.1+3\sqrt{5}+1}\)

\(=\sqrt[3]{\left(\sqrt{5}-1\right)^3}+\sqrt[3]{\left(\sqrt{5}+1\right)^3}=\sqrt{5}-1+\sqrt{5}+1=2\sqrt{5}\)

\(\Rightarrow E=\sqrt{5}\)

c/

\(F=\sqrt[3]{182+25\sqrt{53}}+\sqrt[3]{182-25\sqrt{53}}\)

\(F^3=364+3F\sqrt[3]{182^2-33125}=364-3F\)

\(\Leftrightarrow F^3+3F-364=0\)

\(\Leftrightarrow\left(F-7\right)\left(F^2+7F+52\right)=0\)

\(\Rightarrow F=7\)

Bài 2:

a/ \(C=\frac{\sqrt{2}-1}{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}+\frac{\sqrt{3}-\sqrt{2}}{\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}+\sqrt{2}\right)}+\frac{\sqrt{4}-\sqrt{3}}{\left(\sqrt{4}-\sqrt{3}\right)\left(\sqrt{4}+\sqrt{3}\right)}\)

\(=\sqrt{2}-1+\sqrt{3}-\sqrt{2}+\sqrt{4}-\sqrt{3}\)

\(=\sqrt{4}-1=2-1=1\)

Bài 2

b/

\(D=\frac{\sqrt{3}\left(\sqrt{3}+2\right)}{\sqrt{3}}+\frac{\sqrt{2}\left(\sqrt{2}+1\right)}{\sqrt{2}+1}-\frac{2+\sqrt{3}}{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}\)

\(=\sqrt{3}+2+\sqrt{2}-2-\sqrt{3}\)

\(=\sqrt{2}\)

c/

\(E=\frac{\left(\sqrt{3}-x\right)\left(\sqrt{3}+x\right)}{x+\sqrt{3}}=\sqrt{3}-x\)

d/

\(F=\frac{\left(\sqrt{2}-1\right)}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)}+\frac{\sqrt{3}-\sqrt{2}}{\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}+\sqrt{2}\right)}+...+\frac{\sqrt{2020}-\sqrt{2019}}{\left(\sqrt{2020}-\sqrt{2019}\right)\left(\sqrt{2020}+\sqrt{2019}\right)}\)

\(=\sqrt{2}-1+\sqrt{3}-\sqrt{2}+...+\sqrt{2020}-\sqrt{2019}\)

\(=\sqrt{2020}-1\)

e/

\(G=\sqrt{2+\sqrt{2+\sqrt{2+...}}}\) (G>0)

\(\Rightarrow G^2=2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}\)

\(\Rightarrow G^2=2+G\)

\(\Rightarrow G^2-G-2=0\Rightarrow\left(G+1\right)\left(G-2\right)=0\)

\(\Rightarrow G=2\)