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a)
Có: \(2>1>0\)
\(\Rightarrow\sqrt{2}>1\Rightarrow1+\sqrt{2}>1+1\\ \Leftrightarrow1+\sqrt{2}>2\)
b) Có: \(0< \sqrt{3}< 3\)
\(\Rightarrow3+1>\sqrt{3}+1\\ \Rightarrow4>\sqrt{3}+1\)
c) Có: \(0< \sqrt{11}< \sqrt{25}\left(0< 11< 25\right)\)
\(\Rightarrow\sqrt{11}< 5\\ \Rightarrow-2\sqrt{11}>-2.5=-10\left(-2< 0\right)\)
d) Có: \(0< \sqrt{11}< \sqrt{16}=4\left(do.0< 11< 16\right)\)
\(\Rightarrow3\sqrt{11}< 3.4\\ \Leftrightarrow3\sqrt{11}< 12\)
a: 2=1+1<1+căn 2
b: 4=1+3>1+căn 3
c: -2căn 11=-căn 44
-10=-căn 100
mà 44<100
nên -2 căn 11>-10
d: 12=3*4=3*căn 16>3*căn 11
a: \(1< \sqrt{2}\)
nên \(2< \sqrt{2}+1\)
b: \(2\sqrt{31}=\sqrt{124}\)
\(10=\sqrt{100}\)
mà 124>100
nên \(2\sqrt{31}>10\)
c: \(-3\sqrt{11}=-\sqrt{99}\)
\(-\sqrt{12}=-\sqrt{12}\)
mà 99>12
nên \(-3\sqrt{11}< -\sqrt{12}\)
Lời giải:
\(2\sqrt{12}>2\sqrt{9}=2.3=6>3\)
\(\sqrt{6}> \sqrt{5}\)
\(\Rightarrow 2\sqrt{12}+\sqrt{6}> 3+\sqrt{5}\)
\(A=\left(\sqrt{1}+\sqrt{2}+\sqrt{3}+\sqrt{4}\right)+\left(\sqrt{5}+\sqrt{6}+\sqrt{7}+\sqrt{8}+\sqrt{9}\right)+\left(\sqrt{10}+\sqrt{11}+\sqrt{12}\right)\)
Ta có:
\(\sqrt{1}+\sqrt{2}+\sqrt{3}+\sqrt{4}>1+\sqrt{1}+\sqrt{1}+\sqrt{1}+2=5\)
\(\sqrt{5}+\sqrt{6}+\sqrt{7}+\sqrt{8}+\sqrt{9}>\sqrt{5}+\sqrt{5}+\sqrt{5}+\sqrt{5}+\sqrt{5}=5\sqrt{5}\)
\(\sqrt{10}+\sqrt{11}+\sqrt{12}>\sqrt{9}+\sqrt{9}+\sqrt{9}=9\)
=> \(A>5+5\sqrt{5}+9=14+5\sqrt{5}>12+5\sqrt{5}\)
Vậy...
Ta có: \(12>9\)
\(6\sqrt{3}>4\sqrt{5}\)
Do đó: \(12+6\sqrt{3}>9+4\sqrt{5}\)
\(\Leftrightarrow\sqrt{12+6\sqrt{3}}>\sqrt{9+4\sqrt{5}}\)
\(A=\sqrt{12+\sqrt{12+\sqrt{12}}}+\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6}}}}< \sqrt{12+\sqrt{12+\sqrt{16}}}+\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{9}}}}\)\(=7\)
\(B=\sqrt{14}+\sqrt{11}>\sqrt{13,69}+\sqrt{10,89}=7\)
\(\Rightarrow A< B\)
Ta có:
\(12< 16\Rightarrow\sqrt{12}< \sqrt{16}=4\\ 6< 9\Rightarrow\sqrt{6}< \sqrt{9}=3\)
\(\Rightarrow A< \sqrt{12+\sqrt{12+4}}+\sqrt{6+\sqrt{6+\sqrt{6+3}}}=\sqrt{12+4}+\sqrt{6+3}=4+3=7\) (1)
Lại có :
\(B=\sqrt{14}+\sqrt{11}\Rightarrow B^2=25+2\sqrt{14.11}=25+2\sqrt{154}>25+2\sqrt{144}=25+2.12=49=7^2\)
Mà B > 0
\(\Rightarrow B>7\) (2)
Từ (1),(2) suy ra A<B
a) Có \(\sqrt{2}< \sqrt{2,25}=1,5\)
\(\sqrt{6}< \sqrt{6,25}=2,5\);
\(\sqrt{12}< \sqrt{12,25}=3,5\);
\(\sqrt{20}< \sqrt{20,25}=4,5\)
=> \(P=\sqrt{2}+\sqrt{6}+\sqrt{12}+\sqrt{20}< 1,5+2,5+3,5+4,5=12\)
Vậy P < 12
Answer:
ý a, tham khảo bài làm của @xyzquynhdi
\(\sqrt{2}+\sqrt{3}+\sqrt{5}\)
\(\sqrt{10+\sqrt{24}+\sqrt{40}+\sqrt{60}}\)
\(=\sqrt{10+2\sqrt{6}+2\sqrt{10}+2\sqrt{15}}\)
\(=\sqrt{\left(\sqrt{2}\right)^2+\left(\sqrt{3}\right)^2+\left(\sqrt{5}\right)^2+2\sqrt{2}\sqrt{3}+2\sqrt{2}\sqrt{5}+2\sqrt{3}\sqrt{5}}\)
\(=\sqrt{\left(\sqrt{2}+\sqrt{3}+\sqrt{5}\right)^2}=\sqrt{2}+\sqrt{3}+\sqrt{5}\)
a, \(3\sqrt{3}\) >\(2\sqrt{3}\) =>\(3\sqrt{3}\) >\(\sqrt{12}\)
b,có \(3\sqrt{5}=\sqrt{45}\) <\(\sqrt{49}=7\) =>7 >\(3\sqrt{5}\)
c,\(\sqrt{\dfrac{51}{9}}\) <\(\sqrt{6}\) => \(\dfrac{1}{3}\sqrt{51}\) <\(\dfrac{1}{5}\sqrt{150}\)
d.\(\dfrac{1}{2}\sqrt{6}< 6\sqrt{\dfrac{1}{2}}\)
Đưa thừa số vào trong dấu căn rồi so sánh.
a) 3√3 > √12
b) 7 > 3√5
c)
d)