ta có ;\(36-16\sqrt{5}=16-2\cdot4\cdot2\sqrt{5}+20=\left(2\sqrt{5}-4\right)^2\)
\(12+2\sqrt{35}=7+2\sqrt{7}\cdot\sqrt{5}+5=\left(\sqrt{7}+\sqrt{5}\right)^2\)
\(81-36\sqrt{5}=36-2\cdot6\cdot3\sqrt{5}+45=\left(3\sqrt{5}-6\right)^2\)
\(11+4\sqrt{7}=\sqrt{7}+2\cdot2\cdot\sqrt{7}+4=\left(\sqrt{7}+2\right)^2\)
TỪ ĐÓ TÍNH RA
Cho A =\(\frac{\sqrt{2}-\sqrt{1}}{2+1}+\frac{\sqrt{3}-\sqrt{2}}{3+2}+...+\frac{\sqrt{36}-\sqrt{35}}{36+35}\)
CMR; A<5/12
CMR:
\(A=\frac{\sqrt{2}-1}{2+1}+\frac{\sqrt{3}-2}{3+2}+...+\frac{\sqrt{36}-35}{36+35}< \frac{5}{12}\)
Cho A=\(\frac{\sqrt{2}-\sqrt{1}}{2+1}\)+\(\frac{\sqrt{3}-\sqrt{2}}{3+2}\)+...+\(\frac{\sqrt{36}-\sqrt{35}}{36+35}\)
Chứng minh: A<\(\frac{5}{12}\)
\(A=\frac{\sqrt{2}-\sqrt{1}}{2+1}+\frac{\sqrt{3}-\sqrt{2}}{3+2}+...+\frac{\sqrt{36}-\sqrt{35}}{36+35}\)
chứng minh A < 5/2
a) \(\dfrac{2}{5}\sqrt{25}\) -\(\dfrac{1}{2}\sqrt{4}\) b)0,5\(\sqrt{0,09}\) +5\(\sqrt{0,81}\) c)\(\dfrac{2}{5}\sqrt{\dfrac{25}{36}}\) -\(\dfrac{5}{2}\sqrt{\dfrac{4}{25}}\)
d)-2\(\sqrt{\dfrac{-36}{-16}}\) + 5\(\sqrt{\dfrac{-81}{-25}}\)
a) \(\frac{2\sqrt{5}-4\sqrt{10}}{3\sqrt{10}}\)
b)\(\frac{6\sqrt{6}-2\sqrt{12}+3-\sqrt{2}}{2\sqrt{6}+1}\)
c)\(\frac{\sqrt{3\left(3-\sqrt{11}\right)^2}}{\sqrt{6}\left(3-\sqrt{11}\right)}\)
d)\(\frac{5\sqrt{7}-4\sqrt{35}+7\sqrt{5}}{\sqrt{35}}\)
e) \(\left(\sqrt{3}-1\right)\sqrt{2\sqrt{19+8\sqrt{3}-4}}\)
g) \(\sqrt{47+\sqrt{5}}.\sqrt{7-\sqrt{2+\sqrt{5}}.}\sqrt{7+\sqrt{2+\sqrt{5}}}\)
THỰC HIỆN PHÉP TÍNH
1,\(\sqrt{3+\sqrt{5}}.\sqrt{2}\)
2,\(\sqrt{3-\sqrt{5}.\sqrt{8}}\)
3,\((\sqrt{\dfrac{3}{4}}-\sqrt{3}+5\sqrt{\dfrac{4}{3})}.\sqrt{12}\)
4,\((\sqrt{\dfrac{1}{7}}-\sqrt{\dfrac{16}{7}}+\sqrt{7}):\sqrt{7}\)
5, \(\sqrt{36-12\sqrt{5}}:\sqrt{6}\)
6,\(\sqrt{3-\sqrt{5}:}\sqrt{2}\)
bài 5
ĐK:\(x>2,y>1\)
\(\frac{36}{\sqrt{x-2}}+\frac{4}{\sqrt{y-1}}=28-4\sqrt{x-2}-\sqrt{y-1}..\)\(\Leftrightarrow\frac{36}{\sqrt{x-2}}+4\sqrt{x-2}+\frac{4}{\sqrt{y-1}}+\sqrt{y-1}=28\)
Áp dụng AM-GM ta có:
\(\frac{36}{\sqrt{x-2}}+4\sqrt{x-2}\ge2\sqrt{\frac{144\sqrt{x-2}}{\sqrt{x-2}}}=24\)
\(\frac{4}{\sqrt{y-1}}+\sqrt{y-1}\ge2\sqrt{\frac{4\sqrt{y-1}}{\sqrt{y-1}}}=4.\)
\(\Rightarrow\frac{36}{\sqrt{x-2}}+4\sqrt{x-2}+\frac{4}{\sqrt{y-1}}+\sqrt{y-1}\ge28.\)
Dấu \(=\)xảy ra khi \(\frac{36}{\sqrt{x-2}}=4\sqrt{x-2}\Leftrightarrow x=11\left(n\right).\)
\(\frac{4}{\sqrt{y-1}}=\sqrt{y-1}\Leftrightarrow y=5\left(n\right).\)
Vậy \(x=11,y=5\)
Thực hiện phép tính:
a) \(\sqrt{36}.\sqrt{121}+\sqrt[3]{-64}-\sqrt[3]{125}\)
b) \(\sqrt{75}+\sqrt{\left(\sqrt{3}-2\right)^2}-30\sqrt{\frac{3}{25}}\)
c) \(\sqrt{11-4\sqrt{7}}-\frac{12}{1+\sqrt{7}}\)