Tính ( k dùng máy tính ):
a)\(\sqrt{7}\)
b) \(\sqrt{55}\)
c) \(\sqrt{11}\)
K
Khách
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12 tháng 2 2022
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 \(\sqrt{2}+\sqrt{11}< 5+\sqrt{3}\)
HT
26 tháng 8 2016
a) \(9=6+3=6+\sqrt{9}\)
\(6+2\sqrt{2}=6+\sqrt{8}\)
\(\sqrt{8}< \sqrt{9}\) nên \(6+\sqrt{8}=6+2\sqrt{2}< 6+\sqrt{9}=9\)
b) \(\left(\sqrt{2}+\sqrt{3}\right)^2=5+2\sqrt{6}=5+\sqrt{24}\)
\(3^2=9=5+4=5+\sqrt{16}\)
\(\sqrt{16}< \sqrt{24}\Rightarrow3^2< \left(\sqrt{2}+\sqrt{3}\right)^2\Rightarrow3< \sqrt{2}+\sqrt{3}\)
c) \(9+4\sqrt{5}=\left(2+\sqrt{5}\right)^2\)
\(16=\left(2+2\right)^2=\left(2+\sqrt{4}\right)^2\)
\(\sqrt{4}< \sqrt{5}\Rightarrow2+\sqrt{4}< 2+\sqrt{5}\Rightarrow\left(2+\sqrt{4}\right)^2=16< \left(2+\sqrt{5}\right)^2=9+4\sqrt{5}\)
d) \(\left(\sqrt{11}-\sqrt{3}\right)^2=14-2\sqrt{33}=14-\sqrt{132}\)
\(2^2=14-10=14-\sqrt{100}\)
\(\sqrt{100}< \sqrt{132}\Leftrightarrow-\sqrt{100}>-\sqrt{132}\Leftrightarrow14-\sqrt{100}>14-\sqrt{132}\)
\(\Rightarrow2>\sqrt{11}-\sqrt{3}\)
19 tháng 9 2021
\(a,\left(\sqrt{\sqrt{3}}\right)^4=3< 4=\left(\sqrt{2}\right)^4\Rightarrow\sqrt{\sqrt{3}}< \sqrt{2}\\ b,\left(\sqrt{2\sqrt{3}}\right)^4=12< 18=\left(\sqrt{3\sqrt{2}}\right)^4\Rightarrow\sqrt{2\sqrt{3}}=\sqrt{3\sqrt{2}}\\ c,\left(2+\sqrt{6}\right)^2=8+4\sqrt{6};5^2=25=8+17;\left(4\sqrt{6}\right)^2=96< 289=17^2\\ \Rightarrow4\sqrt{6}< 17\Rightarrow2+\sqrt{6}< 5\\ d,\left(7-2\sqrt{2}\right)^2=57-28\sqrt{2};4^2=16=57-41;\left(28\sqrt{2}\right)^2=1568< 41^2=1681\\ \Rightarrow28\sqrt{2}< 41\Rightarrow7-2\sqrt{2}>4\\ e,\left(\sqrt{15}+\sqrt{8}\right)^2=23+4\sqrt{30};7^2=49=23+26;\left(4\sqrt{30}\right)^2=240< 676=26^2\\ \Rightarrow4\sqrt{30}< 26\Rightarrow\sqrt{15}+\sqrt{8}< 7\)
\(f,\left(\sqrt{37}-\sqrt{14}\right)^2=51-2\sqrt{518};\left(6-\sqrt{15}\right)^2=51-12\sqrt{15};\left(2\sqrt{518}\right)^2=2072;\left(12\sqrt{15}\right)^2=2160\\ \Rightarrow2\sqrt{518}< 12\sqrt{15}\Rightarrow\sqrt{37}-\sqrt{14}>6-\sqrt{15}\)
5 tháng 10 2021
b: \(\sqrt{2017}-\sqrt{2016}=\dfrac{1}{\sqrt{2016}+\sqrt{2017}}\)
\(\sqrt{2016}-\sqrt{2015}=\dfrac{1}{\sqrt{2016}+\sqrt{2015}}\)
mà \(\sqrt{2016}+\sqrt{2017}< \sqrt{2016}+\sqrt{2015}\)
nên \(\sqrt{2017}-\sqrt{2016}>\sqrt{2016}-\sqrt{2015}\)
HN
4
30 tháng 10 2018
Ta có :
\(\left(\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}\right)^2=4+\sqrt{7}-2\sqrt{\left(4+\sqrt{7}\right)\left(4-\sqrt{7}\right)}+4-\sqrt{7}\)
\(8-2\sqrt{16-7}=8-2\sqrt{9}=8-2.3=8-6=2\)
\(\Rightarrow\)\(\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}=\sqrt{2}\) ( vì \(\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}>0\) )
\(\Rightarrow\)\(M=\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}-\sqrt{2}=\sqrt{2}-\sqrt{2}=0\)
Vậy \(M=0\)
Chúc bạn học tốt ~
HQ
Hà Quang Minh
Giáo viên
19 tháng 9 2023
\(a)\sqrt {2250} \approx 47,434;\,\,\,\,\,\,b)\sqrt {12} \approx 3,461;\,\,\,\,\,\,\,c)\sqrt 5 \approx 2,236\,\,\,\,\,\,\,\,\,d)\sqrt {624} \approx 24,980\)
NN
6 tháng 9 2020
a) \(\sqrt{7}.\sqrt{55}.\sqrt{35}.\sqrt{11}=\sqrt{7.55.35.11}=\sqrt{7.5.11.5.7.11}=\sqrt{\left(5.7.11\right)^2}\)
\(=5.7.11=385\)
b) \(\frac{\sqrt{144}}{23}:\frac{\sqrt{16}}{23}=\frac{\sqrt{144}}{23}.\frac{23}{\sqrt{16}}=\frac{\sqrt{144}}{\sqrt{16}}=\sqrt{\frac{144}{16}}=\sqrt{9}=3\)
c) \(\frac{\sqrt{5}}{\sqrt{125}}=\sqrt{\frac{5}{125}}=\sqrt{\frac{1}{25}}=\frac{1}{5}\)
d) \(\frac{\sqrt{135}}{\sqrt{15}}=\sqrt{\frac{135}{15}}=\sqrt{9}=3\)
6 tháng 9 2020
a)\(\sqrt{7}.\sqrt{55}.\sqrt{35}.\sqrt{11}=\left(\sqrt{7}.\sqrt{355}\right).\left(\sqrt{35}.\sqrt{11}\right)=\sqrt{385}.\sqrt{385}=385\)
b) \(\frac{\sqrt{144}}{23}:\frac{\sqrt{16}}{23}=\frac{12}{23}.\frac{23}{4}=3\)
c) \(\frac{\sqrt{5}}{\sqrt{125}}=\sqrt{\frac{5}{125}}=\sqrt{\frac{1}{25}}=\frac{1}{\sqrt{5}}=\frac{\sqrt{5}}{5}\)
d) \(\frac{\sqrt{135}}{\sqrt{15}}=\sqrt{\frac{135}{15}}=\sqrt{9}=3\)
19 tháng 9 2021
a) \(1=\sqrt{1}< \sqrt{2}\)
b) \(2=\sqrt{4}>\sqrt{3}\)
c) \(6=\sqrt{36}< \sqrt{41}\)
d) \(7=\sqrt{49}>\sqrt{47}\)
e) \(2=1+1=\sqrt{1}+1< \sqrt{2}+1\)
f) \(1=2-1=\sqrt{4}-1>\sqrt{3}-1\)
g) \(2\sqrt{31}=\sqrt{4.31}=\sqrt{124}>\sqrt{100}=10\)
h) \(\sqrt{3}>0>-\sqrt{12}\)
i) \(5=\sqrt{25}< \sqrt{29}\)
\(\Rightarrow-5>-\sqrt{29}\)
Không dùng máy tính thì dùng bảng :))