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b: Ta có: \(4\sqrt{5}=\sqrt{4^2\cdot5}=\sqrt{80}\)
\(5\sqrt{3}=\sqrt{5^2\cdot3}=\sqrt{75}\)
mà 80>75
nên \(4\sqrt{5}>5\sqrt{3}\)
Bài 1:
Để M có nghĩa thì \(\left\{{}\begin{matrix}x+4\ge0\\2-x\ge0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ge-4\\x\le2\end{matrix}\right.\Leftrightarrow-4\le x\le2\)
Số giá trị nguyên thỏa mãn điều kiện là:
\(\left(2+4\right)+1=7\)
\(A=\dfrac{2}{\sqrt{17}+\sqrt{15}}\) ; \(B=\dfrac{2}{\sqrt{15}+\sqrt{13}}\)
Mà \(\sqrt{17}+\sqrt{15}>\sqrt{15}+\sqrt{13}>0\)
\(\Rightarrow\dfrac{2}{\sqrt{17}+\sqrt{15}}< \dfrac{2}{\sqrt{15}+\sqrt{13}}\)
\(\Rightarrow A< B\)
\(A=\sqrt{17}-\sqrt{15}=\dfrac{2}{\sqrt{17}+\sqrt{15}}\)
\(B=\sqrt{15}-\sqrt{13}=\dfrac{2}{\sqrt{13}+\sqrt{15}}\)
mà \(\dfrac{2}{\sqrt{17}+\sqrt{15}}< \dfrac{2}{\sqrt{13}+\sqrt{15}}\)
nên A<B
\(a,\left(\sqrt{2}+\sqrt{11}\right)^2=12+2\sqrt{22}\\ \left(\sqrt{3}+5\right)^2=28+10\sqrt{3}\)
Ta thấy \(12< 28;2\sqrt{22}=\sqrt{88}< \sqrt{300}=10\sqrt{3}\)
Nên \(\sqrt{2}+\sqrt{11}< \sqrt{3}+5\)
\(b,\left(\sqrt{21}-\sqrt{5}\right)^2=26-2\sqrt{105}\\ \left(\sqrt{20}-\sqrt{6}\right)^2=26-2\sqrt{120}\)
Vì \(\sqrt{105}< \sqrt{120}\Rightarrow-2\sqrt{105}>-2\sqrt{120}\)
Nên \(\sqrt{21}-\sqrt{5}>\sqrt{20}-\sqrt{6}\)
a: \(\left(\sqrt{7}+\sqrt{15}\right)^2=22+2\sqrt{105}=7+15+2\sqrt{105}\)
\(7^2=49=7+42\)
mà \(15+2\sqrt{105}< 42\)
nên \(\sqrt{7}+\sqrt{15}< 7\)
b: \(\left(\sqrt{2}+\sqrt{11}\right)^2=13+2\sqrt{22}\)
\(\left(5+\sqrt{3}\right)^2=28+10\sqrt{3}=13+15+10\sqrt{3}\)
mà \(2\sqrt{22}< 15+10\sqrt{3}\)
nên \(\sqrt{2}+\sqrt{11}< 5+\sqrt{3}\)
a \(\left(\sqrt{5\sqrt{7}}\right)^4=\left(\left(\sqrt{5\sqrt{7}}\right)^2\right)^2=\left(5\sqrt{7}\right)^2=25\cdot7=175\)
\(=\left(\sqrt{7\sqrt{5}}\right)^4=\left(\left(\sqrt{7\sqrt{5}}\right)^2\right)^2=\left(7\sqrt{5}\right)^2=49\cdot5=240\)
vì 175<240\(\Rightarrow\left(\sqrt{5\sqrt{7}}\right)^4< \left(\sqrt{7\sqrt{5}}\right)^4\Rightarrow\sqrt{5\sqrt{7}}< \sqrt{7\sqrt{5}}\)
b \(6=\sqrt{36}\)
\(\sqrt{31}< \sqrt{36};\sqrt{19}>\sqrt{17}\Rightarrow\sqrt{31}-\sqrt{19}< \sqrt{36}-\sqrt{17}=6-\sqrt{17}\)
c \(\left(\sqrt{10}+\sqrt{17}\right)^2=10+2\sqrt{10\cdot17}+17=27+2\sqrt{170}\)
\(\left(\sqrt{61}\right)^2=61=27+34=27+2\cdot17=27+2\sqrt{289}\)
vì \(2\sqrt{170}< 2\sqrt{289}\Rightarrow27+2\sqrt{170}< 27+2\sqrt{289}\Rightarrow\left(\sqrt{10}+\sqrt{17}\right)^2< \left(\sqrt{61}\right)^2\)
\(\Rightarrow\sqrt{10}+\sqrt{17}< \sqrt{61}\)
Bài này dễ lắm
Câu 1
\(-\sqrt{5}\) lớn hơn \(-2\) . Vì
\(-\sqrt{5}=-2,2236067977\)
\(-2=-2\)
Câu 2
\(\sqrt{2}+\sqrt{3}\) bé hơn \(\sqrt{10}\) . Vì
\(\sqrt{2}+\sqrt{3}=3,146264\)
\(\sqrt{10}=3,16227766\)
Câu 3
\(8\) lớn hơn \(\sqrt{15}+\sqrt{17}\)
\(8=8\)
\(\sqrt{15}+\sqrt{17}=7,996088972\)
a: \(\left(\sqrt{3}+\sqrt{5}\right)^2=8+\sqrt{60}\)
\(\left(\sqrt{17}\right)^2=17=8+\sqrt{81}\)
mà 60<81
nên \(3+\sqrt{5}< \sqrt{17}\)
c: \(\left(\sqrt{2004}+\sqrt{2006}\right)^2=4010+2\cdot\sqrt{2005^2-1}\)
\(\left(2\cdot\sqrt{2005}\right)^2=8020=4010+2\cdot\sqrt{2005^2}\)
mà \(2005^2-1< 2005^2\)
nên \(\sqrt{2004}+\sqrt{2006}< 2\sqrt{2005}\)
d: \(\left(\sqrt{5}+2\right)^2=9+4\sqrt{5}=9+\sqrt{80}\)
\(\left(\sqrt{3}+\sqrt{6}\right)^2=9+2\cdot\sqrt{3\cdot6}=9+\sqrt{72}\)
mà 80>72
nên \(\sqrt{5}+2>\sqrt{3}+\sqrt{6}\)