\(B=\sqrt{\left(3-\sqrt{7}\right)^2+\sqrt{11+4\sqrt{7}}}\)
\(=\sqrt{9-6\sqrt{7}+7+\sqrt{\left(2+\sqrt{7}\right)^2}}\)
\(=\sqrt{16-6\sqrt{7}+2+\sqrt{7}}\)
\(=\sqrt{18-5\sqrt{7}}\)
B = \(\sqrt{\left(3-\sqrt{7}\right)^2}\) + \(\sqrt{11+4\sqrt{7}}\)
Chứng minh rằng:
\(\dfrac{1}{3\left(\sqrt{2}+1\right)}+\dfrac{1}{5\left(\sqrt{3}+\sqrt{2}\right)}+\dfrac{1}{7\left(\sqrt{4}+\sqrt{3}\right)}+...+\dfrac{1}{4021\left(\sqrt{2011}+\sqrt{2010}\right)}< \dfrac{1}{2}\left(1-\dfrac{1}{\sqrt{2011}}\right)\)
Rút gọn biểu thức
M = \(\dfrac{2}{\sqrt{7}-\sqrt{6}}-\sqrt{28}+\sqrt{54}\)
N= \(\left(2-\sqrt{3}\right)\sqrt{26+15\sqrt{3}}-\left(2+\sqrt{3}\right)\sqrt{26-15\sqrt{3}}\)
giải phương trình vô tỉ
1,\(\sqrt{1-\sqrt{x}}+\sqrt{4+x}=3\)
2,\(\sqrt{x+1}+\sqrt[3]{7-x}=2\)
3,\(\sqrt{x}+\sqrt{x+1}=\sqrt{x-1}+\sqrt{x+4}\)
4,\(\left(x+3\right)\sqrt{10-x^2}=x^2-x-12\)
5,\(\sqrt[3]{x-1}+\sqrt[3]{x-2}=\sqrt[3]{2x-3}\)
Biểu thức \(\sqrt{\left(\sqrt{7}-5\right)^2}\)+\(\sqrt{\left(2-\sqrt{7}\right)^2}\) có giá trị bằng
cho biểu thức P = \(\left(\dfrac{1-a\sqrt{a}}{1-\sqrt{a}}+\sqrt{a}\right).\left(\dfrac{1+a\sqrt{a}}{1+\sqrt{a}}-\sqrt{a}\right)\)
a, rút gọn P
b, tìm a để P < \(7-4\sqrt{3}\)
thực hiện phép tính
a, \(\dfrac{\sqrt{3-\sqrt{5}}.\left(3+\sqrt{5}\right)}{\sqrt{10}+\sqrt{2}}\)
b, \(\sqrt{8\sqrt{3}}-2\sqrt{25\sqrt{12}}+4\sqrt{\sqrt{192}}\)
c, \(\sqrt{2-\sqrt{3}}.\left(\sqrt{5}+\sqrt{2}\right)\)
d, \(\sqrt{3-\sqrt{5}}+\sqrt{3+\sqrt{5}}\)
e, \(\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\)
f, \(\left(5+2\sqrt{6}\right)\left(49-20\sqrt{6}\right)\sqrt{5-2\sqrt{6}}\)
g, \(\dfrac{1}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\dfrac{1}{\sqrt{2}-\sqrt{2-\sqrt{3}}}\)
h, \(\dfrac{6+4\sqrt{2}}{\sqrt{2}+\sqrt{6+4\sqrt{2}}}+\dfrac{6-4\sqrt{2}}{\sqrt{2}+\sqrt{6+4\sqrt{2}}}\)
i, \(\dfrac{\left(\sqrt{5+2}\right)^2-8\sqrt{5}}{2\sqrt{5}-4}\)
k, \(\sqrt{14-8\sqrt{3}}-\sqrt{24-12\sqrt{3}}\)
l, \(\dfrac{4}{\sqrt{3}+1}+\dfrac{1}{\sqrt{3}-2}+\dfrac{6}{\sqrt{3}-3}\)
m, \(\left(\sqrt{2}+1\right)^3-\left(\sqrt{2}-1\right)^3\)
n, \(\dfrac{\sqrt{3}}{1-\sqrt{\sqrt{3+1}}}+\dfrac{\sqrt{3}}{1+\sqrt{\sqrt{3+1}}}\)
A=\(\left(\dfrac{x-5\sqrt{x}+4}{x\sqrt{x}-3x+2\sqrt{x}}-\dfrac{3\sqrt{x}+3}{\sqrt{x}+2-x}\right):\left(\dfrac{x-\sqrt{x}-6}{x-3\sqrt{x}}-\dfrac{x-2\sqrt{x}}{x-4\sqrt{x}+4}\right)+\sqrt{x}\)a). Rút gọn A
b). Cho a,b là 2 số dương thỏa mãn a+b≥4. tìm GTNN của biểu thức B=\(5a+11b+\dfrac{2}{a}+\dfrac{72}{b}\)
Giải hệ phương trình:
a) \(\left\{{}\begin{matrix}\sqrt{3y^2+13}-\sqrt{15-2x}=\sqrt{x+1}\\y^4-2x^2y+7y^2=\left(x+1\right)\left(8-x\right)\end{matrix}\right.\)
b) \(\left\{{}\begin{matrix}\sqrt{x+y}-\sqrt{x-y}=2\\\sqrt{x^2+y^2+1}-\sqrt{x^2-y^2}=3\end{matrix}\right.\)
c) \(\left\{{}\begin{matrix}\sqrt{2x+y+1}-\sqrt{x+y}=3\\\sqrt{3\left(x+y\right)^2+1}+\sqrt{x-5}=5\end{matrix}\right.\)
