tính giá trị \(\sqrt{6+3\sqrt{3}}-\sqrt{6-3\sqrt{3}}\)
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24 tháng 8 2021
Đặt \(\sqrt[3]{6-2\sqrt{7}}=a\), \(\sqrt[3]{6+2\sqrt{7}}=b\)
\(\Rightarrow\left\{{}\begin{matrix}a^3+b^3=12\\ab=2\end{matrix}\right.\)
\(x=\sqrt[3]{6-2\sqrt{7}}+\sqrt[3]{6+2\sqrt{7}}=a+b\)
\(\Rightarrow x^3=a^3+b^3+3ab\left(a+b\right)=12+3.2\left(a+b\right)=12+6x\)
\(\Rightarrow x^3-6x-12=0\)
\(Q=x^3-6x+17=\left(x^3-6x-12\right)+29=29\)
30 tháng 8 2021
\(13-\sqrt{\left(\sqrt{2}-3\right)^2}\)
=13-/\(\sqrt{2}\)-3/
=13-(3-\(\sqrt{2}\))
=13-3+\(\sqrt{2}\)
=10\(\sqrt{2}\)
\(\sqrt{3-2\sqrt{2}}-\sqrt{6-4\sqrt{2}}\)
\(=\sqrt{2-2\sqrt{2}+1}-\sqrt{4-2.2\sqrt{2}+2}\)
\(=\sqrt{\left(\sqrt{2}-1\right)^2}-\sqrt{\left(2-\sqrt{2}\right)^2}\)
\(=\sqrt{2}-1-2+\sqrt{2}\)
\(=2\sqrt{2}-3\)
30 tháng 8 2021
1) \(13-\sqrt{\left(\sqrt{2}-3\right)^2}=13-\left|\sqrt{2}-3\right|=13+\sqrt{2}-3=10+\sqrt{2}\)
2) \(\sqrt{3-2\sqrt{2}}-\sqrt{6-4\sqrt{2}}=\sqrt{\left(\sqrt{2}-1\right)^2}-\sqrt{\left(2-\sqrt{2}\right)^2}=\left|\sqrt{2}-1\right|-\left|2-\sqrt{2}\right|=\sqrt{2}-1-2+\sqrt{2}=2\sqrt{2}-3\)
2 tháng 7 2021
a) Ta có: \(\left(7\sqrt{48}+3\sqrt{27}-2\sqrt{12}\right)\cdot\sqrt{3}\)
\(=\left(7\cdot4\sqrt{3}+3\cdot3\sqrt{3}-2\cdot2\sqrt{3}\right)\cdot\sqrt{3}\)
\(=33\sqrt{3}\cdot\sqrt{3}\)
=99
b) Ta có: \(\left(12\sqrt{50}-8\sqrt{200}+7\sqrt{450}\right):\sqrt{10}\)
\(=\left(12\cdot5\sqrt{2}-8\cdot10\sqrt{2}+7\cdot15\sqrt{2}\right):\sqrt{10}\)
\(=\dfrac{85\sqrt{2}}{\sqrt{10}}=\dfrac{85}{\sqrt{5}}=17\sqrt{5}\)
c) Ta có: \(\left(2\sqrt{6}-4\sqrt{3}+5\sqrt{2}-\dfrac{1}{4}\sqrt{8}\right)\cdot3\sqrt{6}\)
\(=\left(2\sqrt{6}-4\sqrt{3}+5\sqrt{2}-\dfrac{1}{4}\cdot2\sqrt{2}\right)\cdot3\sqrt{6}\)
\(=\left(2\sqrt{6}-4\sqrt{3}+3\sqrt{2}\right)\cdot3\sqrt{6}\)
\(=36-36\sqrt{2}+18\sqrt{3}\)
d) Ta có: \(3\sqrt{15\sqrt{50}}+5\sqrt{24\sqrt{8}}-4\sqrt{12\sqrt{32}}\)
\(=3\cdot\sqrt{75\sqrt{2}}+5\cdot\sqrt{48\sqrt{2}}-4\sqrt{48\sqrt{2}}\)
\(=3\cdot5\sqrt{2}\cdot\sqrt{\sqrt{2}}+4\sqrt{3}\sqrt{\sqrt{2}}\)
\(=15\sqrt{\sqrt{8}}+4\sqrt{\sqrt{18}}\)
2 tháng 7 2021
a,=\(\left(28\sqrt{3}+9\sqrt{3}-4\sqrt{3}\right).\sqrt{3}\)
\(=28.3+9.3-4.3=99\)
b,\(=\left(60\sqrt{2}-80\sqrt{2}+175\sqrt{2}\right):\sqrt{10}\)
\(=155\sqrt{2}:\sqrt{10}=\dfrac{155}{\sqrt{5}}\)
AH
Akai Haruma
Giáo viên
17 tháng 9 2021
Lời giải:
\(\sqrt{2}A=\sqrt{4+2\sqrt{3}}-\sqrt{4-2\sqrt{3}}=\sqrt{(\sqrt{3}+1)^2}-\sqrt{(\sqrt{3}-1)^2}\)
\(=|\sqrt{3}+1|-|\sqrt{3}-1|=\sqrt{3}+1-(\sqrt{3}-1)=2\)
$\Rightarrow A\geq \sqrt{2}$
\(B=2\sqrt{6}-4\sqrt{2}+(9+4\sqrt{2})-2\sqrt{6}=2\sqrt{6}-4\sqrt{2}+9+4\sqrt{2}-2\sqrt{6}\)
\(=9\)
17 tháng 9 2021
a)ta có:\(A^2=\left(\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}\right)\)=\(2+\sqrt{3}+2-\sqrt{3}-2\sqrt{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}\)
=\(4-2\sqrt{1}=4-2=2\)
\(\Rightarrow A=\pm\sqrt{2}\) mà A>0\(\Rightarrow A=\sqrt{2}\)
b)B=\(2\sqrt{6}-4\sqrt{2}+1+4\sqrt{2}+8-2\sqrt{6}\)=9
11 tháng 8 2017
ai nay dung kinh nghiem la chinh
cau a)
ta thay \(10+6\sqrt{3}=\left(1+\sqrt{3}\right)^3\)
\(6+2\sqrt{5}=\left(1+\sqrt{5}\right)^2\)
khi do \(x=\frac{\sqrt[3]{\left(\sqrt{3}+1\right)^3}\left(\sqrt{3}-1\right)}{\sqrt{\left(1+\sqrt{5}\right)^2}-\sqrt{5}}\)
\(x=\frac{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}{1+\sqrt{5}-\sqrt{5}}\)
\(x=\frac{3-1}{1}=2\)
suy ra
x^3-4x+1=1
A=1^2018
A=1
b)
ta thay
\(7+5\sqrt{2}=\left(1+\sqrt{2}\right)^3\)
khi do
\(x=\sqrt[3]{\left(1+\sqrt{2}\right)^3}-\frac{1}{\sqrt[3]{\left(1+\sqrt{2}\right)^3}}\)
\(x=1+\sqrt{2}-\frac{1}{1+\sqrt{2}}=\frac{\left(1+\sqrt{2}\right)^2-1}{1+\sqrt{2}}=\frac{2+2\sqrt{2}}{1+\sqrt{2}}\)
x=2
thay vao
x^3+3x-14=0
B=0^2018
B=0
26 tháng 6 2023
Giải
Ta có:
\(x=\sqrt{2+\sqrt{2+\sqrt{3}}-\sqrt{6-3\sqrt{2+\sqrt{3}}}}\)
Khi đó:
\(x^2=\left(\sqrt{2+\sqrt{2+\sqrt{3}}-\sqrt{6-3\sqrt{2+\sqrt{3}}}}\right)^2\\ =2+\sqrt{2+\sqrt{3}}+6-3\sqrt{2+\sqrt{3}}-2\sqrt{\left(2+\sqrt{2+\sqrt{3}}\right)\left(6-3\sqrt{2+\sqrt{3}}\right)}\\ =8-2\sqrt{2+\sqrt{3}}-2\sqrt{12-3\left(2+\sqrt{3}\right)}\\ =8-\sqrt{2}.\sqrt{4+2\sqrt{3}}-2\sqrt{6-3\sqrt{3}}\\ =8-\sqrt{2}.\sqrt{4+2\sqrt{3}}-\sqrt{2}.\sqrt{12-6\sqrt{3}}\\ =8-\sqrt{2}.\left(\sqrt{4+2\sqrt{3}}+\sqrt{12-6\sqrt{3}}\right)\\ =8-\sqrt{2}.\left(\sqrt{\left(\sqrt{3}\right)^2+2\sqrt{3}+1}+\sqrt{9-2.3\sqrt{3}+\left(\sqrt{3}\right)^2}\right)\\ 8-\sqrt{2}.\left(\sqrt{\left(\sqrt{3}+1\right)^2}+\sqrt{\left(3-\sqrt{3}\right)^2}\right)\\ =8-\sqrt{2}.\left(\sqrt{3}+1+3-\sqrt{3}\right)\\ =8-4\sqrt{2}\\ \Rightarrow x^4-16x^2=\left(8-4\sqrt{2}\right)^2-16.\left(8-4\sqrt{2}\right)\\ =96-64\sqrt{2}-128+64\sqrt{2}=-32\)
Vậy \(S=-32\)
NB
6 tháng 3 2022
\(=\left(4\sqrt{3}-2\sqrt{2}\right)\left(\sqrt{12+4\sqrt{6}+2+8\sqrt{3}+4\sqrt{2}+4-2}\right)\\ =\left(4\sqrt{3}-2\sqrt{2}\right)\left(\sqrt{\left(2\sqrt{3}+\sqrt{2}\right)^2+4\left(2\sqrt{3}+\sqrt{2}\right)+4-2}\right)\\ =\left(4\sqrt{3}-2\sqrt{2}\right)\left(\sqrt{\left(2\sqrt{3}+\sqrt{2}+2\right)^2-2}\right)\\ =\left(4\sqrt{3}-2\sqrt{2}\right)\left(2\sqrt{3}+\sqrt{2}\right)=20\)
\(A=\sqrt{6+3\sqrt{3}}-\sqrt{6-3\sqrt{3}}.\)
Do đó : \(A.\sqrt{6+3\sqrt{3}}=\sqrt{6+3\sqrt{3}}\left(\sqrt{6+3\sqrt{3}}-\sqrt{6-3\sqrt{3}}\right)=3+3\sqrt{3}.\) (1)
\(A.\sqrt{6-3\sqrt{3}}=\sqrt{6-3\sqrt{3}}\left(\sqrt{6+3\sqrt{3}}-\sqrt{6-3\sqrt{3}}\right)=3\sqrt{3}-3.\) (2)
Nhân các đẳng thức (1) và (2) vế theo vế tương ứng, được :
\(3A^2=\left(3\sqrt{3}+3\right)\left(3\sqrt{3}-3\right)\Leftrightarrow3A^2=18\Leftrightarrow A^2=6\Rightarrow A=\sqrt{6}.\)
Vậy : \(A=\sqrt{6+3\sqrt{3}}-\sqrt{6-3\sqrt{3}}=\sqrt{6}.\)
\(\sqrt{6+3\sqrt{3}}-\sqrt{6-3\sqrt{3}}\)
\(=\sqrt{\left(\sqrt{6+3\sqrt{3}}-\sqrt{6-3\sqrt{3}}\right)^2}\)
\(=\sqrt{6+3\sqrt{3}+6-3\sqrt{3}-2.\sqrt{6+3\sqrt{3}}.\sqrt{6-3\sqrt{3}}}\)
\(=\sqrt{12-6}=\sqrt{6}\)