1. Tính giá trị biểu thức: \(A=\sqrt{a^2+4ab^2+4b}-\sqrt{4a^2-12ab^2+9b^4}\) với \(a=\sqrt{2}\) ; \(b=1\)
2. Đặt \(M=\sqrt{57+40\sqrt{2}}\) ; \(N=\sqrt{57-40\sqrt{2}}\). Tính giá trị của các biểu thức sau:
a) M-N
b) \(M^3-N^3\)
3. Chứng minh: \(\left(\frac{x\sqrt{x}+3\sqrt{3}}{x-\sqrt{3x}+3}-2\sqrt{x}\right)\left(\frac{\sqrt{x}+\sqrt{3}}{3-x}\right)=1\) (với \(x\ge0\) và \(x\ne3\))
4. Chứng minh: \(\frac{\left(\sqrt{a}-\sqrt{b}\right)^2+4\sqrt{ab}}{\sqrt{a}+\sqrt{b}}.\frac{a\sqrt{b}-b\sqrt{a}}{\sqrt{ab}}=a-b\) (a > 0 ; b > 0)
5. Chứng minh: \(\sqrt{9+4\sqrt{2}}=2\sqrt{2}+1\) ; \(\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}=5+3\sqrt{2}\) ; \(3-2\sqrt{2}=\left(1-\sqrt{2}\right)^2\)
6. Chứng minh: \(\left(\frac{1}{2\sqrt{2}-\sqrt{7}}-\left(3\sqrt{2}+\sqrt{17}\right)\right)^2=\left(\frac{1}{2\sqrt{2}-\sqrt{17}}-\left(2\sqrt{2}-\sqrt{17}\right)\right)^2\)
7. Chứng minh đẳng thức: \(\left(\frac{3\sqrt{2}-\sqrt{6}}{\sqrt{27}-3}-\frac{\sqrt{150}}{3}\right).\frac{1}{\sqrt{6}}=-\frac{4}{3}\)
8.Chứng minh: \(\frac{2002}{\sqrt{2003}}+\frac{2003}{\sqrt{2002}}>\sqrt{2002}+\sqrt{2003}\)
9. Chứng minh rằng: \(\sqrt{2000}-2\sqrt{2001}+\sqrt{2002}< 0\)
10. \(\frac{1}{2}+\frac{1}{3\sqrt{2}}+...+\frac{1}{\left(n+1\right)\sqrt{n}}< 2\) ; \(\frac{7}{5}< \frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}< \frac{29}{30}\)
1,Trục căn thức ở mẫu, rút gọn: ( với \(x\ge0;x\ne1\))
a,\(\frac{\sqrt{6}+\sqrt{14}}{2\sqrt{3}+\sqrt{28}}\)
b,\(\frac{\sqrt{2}+1}{\sqrt{2}-1}\)
2,Chứng minh các đẳng thức sau:
a,\(\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}=1\)
b,\(\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}}=\sqrt{6}\)
c,\(\left(\frac{\sqrt{a}}{\sqrt{a}+2}+\frac{\sqrt{a}}{\sqrt{a}-2}+\frac{4\sqrt{a}-1}{a-4}\right):\frac{1}{a-4}=-1\)
d,\(\frac{\sqrt{a}+\sqrt{b}}{2\sqrt{a}-2\sqrt{b}}-\frac{\sqrt{a}-\sqrt{b}}{2\sqrt{a}+2\sqrt{b}}-\frac{2b}{b-a}=\frac{2\sqrt{b}}{\sqrt{a}-\sqrt{b}}\)
Chứng minh:\(\sqrt{X^2+\frac{1}{x^2}}+\sqrt{y^2+\frac{1}{y^2}}+\sqrt{z^2+\frac{1}{z^2}}>=\frac{3}{2}\sqrt{17}\)
Bài 1: Chứng minh các đẳng thức sau:
a) \(\sqrt{7-2\sqrt{10}}=\sqrt{5}-\sqrt{2}\)
b)\(\sqrt{4+2\sqrt{3}}-\sqrt{3}=1\)
Bài 2: Thực hiện phép tính:
a) \(\frac{4}{\sqrt{3}+1}\frac{1}{\sqrt{3}-2}+\frac{6}{\sqrt{3}-3}\)
b) \(\frac{4}{3+\sqrt{5}}-\frac{8}{1+\sqrt{5}}+\frac{15}{\sqrt{5}}\)
c) \(\frac{3\sqrt{2}-2\sqrt{3}}{\sqrt{3}-\sqrt{2}}-\frac{5}{\sqrt{6+1}}\)
Chứng minh đẳng thức :
a) \(A=\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}=\sqrt{2}\)
b) \(B=\left(5+\sqrt{21}\right)\left(\sqrt{14}-\sqrt{6}\right)\sqrt{5-\sqrt{21}}=8\)
Chứng minh rằng
\(\sqrt{\frac{2+2\sqrt{3}}{2-2\sqrt{3}}}\) - \(\sqrt{\frac{2-2\sqrt{3}}{2+2\sqrt{3}}}\)= \(2\sqrt{3}\)
a,\(\frac{2\sqrt{2}}{\sqrt{2\sqrt{2}+1}+1}-\frac{2\sqrt{2}}{\sqrt{2\sqrt{2}+1}-1}\)
b,\(\frac{\sqrt{15}-\sqrt{12}}{\sqrt{5}-2}+\frac{1}{2-\sqrt{3}}\)
c,\(\frac{2}{\sqrt{3}-\sqrt{5}}+\frac{3-2\sqrt{3}}{\sqrt{3}-2}\)
d,\(\frac{-4}{\sqrt{7}-\sqrt{5}}+\frac{1}{\sqrt{3}-1}+\frac{4-2\sqrt{5}}{\sqrt{5}-2}\)
e,\(\frac{6}{\sqrt{5}-1}+\frac{7}{1-\sqrt{3}}-\frac{2}{\sqrt{3}-\sqrt{5}}\)
\(\frac{3+2\sqrt{3}}{\sqrt{3}}+\frac{2+\sqrt{2}}{\sqrt{2}+1}-\left(\sqrt{2}+3\right)\)
\(0.1\cdot\sqrt{\left(-3\right)^2}\cdot\left[6\sqrt{\left(\frac{1}{3}\right)^2}-\sqrt{\left(\sqrt{3}-2\right)^2}\right]^2\)
\(\left(\frac{3\sqrt{2}+\sqrt{6}}{\sqrt{12}+2}-\frac{\sqrt{54}}{3}\right)\cdot\frac{2}{\sqrt{6}}\)
\(\left(\frac{3+2\sqrt{3}}{\sqrt{3}+2}+\frac{2+\sqrt{2}}{\sqrt{2}+1}\right)\div\left(1\div\frac{1}{\sqrt{2}+\sqrt{3}}\right)\)
\(\sqrt{\frac{5+2\sqrt{6}}{5-2\sqrt{6}}}+\sqrt{\frac{5-2\sqrt{6}}{5+2\sqrt{6}}}\)
\(\frac{\sqrt{3}+\sqrt{2}-1}{2+\sqrt{6}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+1}\left(\frac{\sqrt{3}}{2-\sqrt{6}}+\frac{\sqrt{3}}{2+\sqrt{6}}\right)-\frac{1}{\sqrt{2}}\)
