Chứng minh : \(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+....+\frac{1}{\sqrt{225}}< 28\)
rút gọn biểu thức
a) \(\frac{3}{2+\sqrt{3}}+\frac{13}{4-\sqrt{3}}+\frac{6}{\sqrt{3}}\)
b) \(\left(\frac{\sqrt{14}-\sqrt{7}}{\sqrt{2}-1}+\frac{\sqrt{15}-\sqrt{5}}{\sqrt{3}-1}\right):\frac{1}{\sqrt{7}-\sqrt{5}}\)
c) \(\sqrt{\left(2+\sqrt{3}\right)^2}-\sqrt{28-10\sqrt{3}}\)
d) \(\frac{3}{3+2\sqrt{3}}+\frac{3}{3-2\sqrt{3}}\)
e) \(\sqrt{20}-15\sqrt{\frac{1}{5}}+\sqrt{\left(1-\sqrt{5}\right)^2}\)
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,\(\frac{\sqrt{a}+\sqrt{b}}{2\sqrt{a}-2\sqrt{b}}-\frac{\sqrt{a}-\sqrt{b}}{2\sqrt{a}+2\sqrt{b}}-\frac{2b}{b-a}=\frac{2\sqrt{b}}{\sqrt{a}-\sqrt{b}}\)
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,\(\frac{6}{\sqrt{5}-1}+\frac{7}{1-\sqrt{3}}-\frac{2}{\sqrt{3}-\sqrt{5}}\)
a) CMR: \(\frac{1}{\sqrt{a+3}+\sqrt{a+2}}+\frac{1}{\sqrt{a+2}+\sqrt{a+1}}+\frac{1}{\sqrt{a+1}+\sqrt{a}}=\frac{3}{\sqrt{a+3}+\sqrt{a}}\)
b) Cho các số thực dương x, y, z thỏa mãn x+y+z=1. CMR: \(\frac{x}{x+yz}+\frac{y}{y+xz}+\frac{z}{z+xy}\le\frac{9}{4}\)
tìm điều kiện để căn thức bậc hai có nghĩa \(\sqrt{\frac{-5}{2x+1}}\)
rút gọn biểu thức
a) \(\sqrt{\left(3-\sqrt{2}\right)^2}+\sqrt{2\left(-5\right)^2}\)
b)\(\frac{\sqrt{6}-\sqrt{3}}{\sqrt{2}-1}-\frac{2}{\sqrt{3}-1}\)
c) \(\frac{\sqrt{8}-2}{\sqrt{2}-1}+\frac{2}{\sqrt{3}-1}-\frac{3}{\sqrt{3}}\)
Thực hiện phép tính
a\(\left(1+\frac{5-\sqrt{5}}{1-\sqrt{5}}\right).\left(\frac{1+\sqrt{5}}{1+\sqrt{5}}+1\right)\) b\(\frac{1}{\sqrt{3}+\sqrt{2}-\sqrt{5}}-\frac{1}{\sqrt{3}+\sqrt{2}+\sqrt{5}}\)
c \(\frac{1}{\sqrt{3}}+\frac{1}{3\sqrt{2}}+\frac{1}{\sqrt{3}}\sqrt{\frac{5}{12}-\frac{1}{\sqrt{6}}}\)
1. Cho A = \(\frac{x-3}{\sqrt{x-1}+\sqrt{2}}\). Tìm GTNN của A
2. Cho B = \(\frac{6-x-\sqrt{x}}{\sqrt{x}+3}\). Tìm GTLN của B
3. Cho C = \(\left(\frac{\sqrt{a}}{2}-\frac{1}{2\sqrt{a}^{ }}\right)\)tất cả bình phương . \(\left(\frac{\sqrt{a}-1}{\sqrt{a}+1}-\frac{\sqrt{a}+1}{\sqrt{a}-1}\right)\). Tìm a để C >0, Tìm a để C = -2
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THỰC HIỆN PHÉP TÍNH:
22) \(\frac{1}{\sqrt{5}+\sqrt{2}}+\frac{1}{\sqrt{5}-\sqrt{2}}\)
23) \(\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}\)
24) \(\frac{\sqrt{18}}{\sqrt{2}}-\frac{\sqrt{12}}{\sqrt{3}}\)
25) \(\sqrt{\left(\sqrt{5}+1\right)^2}+\sqrt{\left(\sqrt{5}-1\right)^2}\)
27) \(\sqrt{3-2\sqrt{2}}\)
28) \(\frac{1}{\sqrt{8}+\sqrt{7}}+\sqrt{175}-2\sqrt{2}\)
30) \(\left(2\sqrt{1\frac{9}{16}}-\sqrt{5\frac{1}{16}}\right):\sqrt{16}\)
34) \(\frac{\left(5\sqrt{3}+\sqrt{50}\right)\left(5-\sqrt{24}\right)}{\sqrt{75}-5\sqrt{2}}\)
35) \(\left(2\sqrt{6}-4\sqrt{3}+5\sqrt{2}-\frac{1}{4}\sqrt{8}\right).3\sqrt{6}\)
36) \(\frac{2\sqrt{8}-\sqrt{12}}{\sqrt{18}-\sqrt{48}}-\frac{\sqrt{5}+\sqrt{27}}{\sqrt{30}+\sqrt{162}}\)
39) \(\sqrt{\frac{2-\sqrt{3}}{2+\sqrt{3}}+\sqrt{\frac{2+\sqrt{3}}{2-\sqrt{3}}}}\)
45) \(\frac{\sqrt{6-2\sqrt{5}}}{2-\sqrt{20}}\)