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a) Có \(\sqrt{25}=5;\sqrt{45}< \sqrt{49}=7\)
\(\Rightarrow\sqrt{25}+\sqrt{45}< 5+7=12\)
Vậy \(\sqrt{25}+\sqrt{45}< 12.\)
b) có \(\left(\sqrt{2013}+\sqrt{2015}\right)^2=2013+2015+2\sqrt{2013}.\sqrt{2015}\)\(=4028+2\sqrt{2013.2015}\)
\(\left(2\sqrt{2014}\right)^2=4.2014=4028+2.2014=4028+2\sqrt{2014^2}\)
Xét \(2014^2-2013.2015=2014.\left(2013+1\right)-2013\left(2014+1\right)\)
\(=2013.2014+2014-2013.2014-2013=1>0\)
\(\Rightarrow2\sqrt{2013.2015}< 2\sqrt{2014^2}\)
Hay \(\left(\sqrt{2013}+\sqrt{2015}\right)^2< \left(2\sqrt{2014}\right)^2\)
\(\Rightarrow\sqrt{2013}+\sqrt{2015}< 2\sqrt{2014}\)
Vậy \(\sqrt{2013}+\sqrt{2015}< 2\sqrt{2014}.\)
c) Có \(\left(\sqrt{2014}-\sqrt{2013}\right)\left(\sqrt{2014}+\sqrt{2013}\right)=2014-2013=1\)\(\rightarrow\sqrt{2014}-\sqrt{2013}=\dfrac{1}{\sqrt{2014}+\sqrt{2013}}\)
Mà \(\sqrt{2014}>\sqrt{2013};\sqrt{2013}>\sqrt{2012}\)
\(\rightarrow\sqrt{2014}+\sqrt{2013}>\sqrt{2013}+\sqrt{2012}\)
Hay \(\dfrac{1}{\sqrt{2014}+\sqrt{2013}}< \dfrac{1}{\sqrt{2013}+\sqrt{2012}}\)
Tương tự, ta có \(\dfrac{1}{\sqrt{2013}+\sqrt{2012}}=\sqrt{2013}-\sqrt{2012}\)
\(\Rightarrow\sqrt{2014}-\sqrt{2013}< \sqrt{2013}-\sqrt{2012}\)
Vậy \(\sqrt{2014}-\sqrt{2013}< \sqrt{2013}-\sqrt{2012}.\)
lop8. thi ap bdt nhu thanh song,
a)
VT=√25+√45<√2(25+45)=√140<√144=12=VP
b)
VT=√2013+√2015<√[2(2013+2015)]=√[4.2014]=2√(2014)=VP.
c) C=VT-VP
√2014+√2012-2√2012
kq(b)=> C<0
VT<VP
So sánh:
\(a,\sqrt{25+9}\)và \(\sqrt{25}+\sqrt{9}\)
Ta có:
\(\sqrt{25+9}=\sqrt{34}< \sqrt{36}=6\) \(\left(1\right)\)
\(\sqrt{25}+\sqrt{9}=\sqrt{5^2}+\sqrt{3^2}=5+3=8\) \(\left(2\right)\)
Từ (1) và (2) \(\Rightarrow\sqrt{25+9}< \sqrt{25}+\sqrt{9}\)
\(b,\sqrt{25-16}\) và \(\sqrt{25}-\sqrt{16}\)
Tương tự:)
a)\(\sqrt{25}+2\sqrt{49}=5+2\cdot7=5+14=19\)
b) \(\sqrt{16}\cdot\sqrt{25}+\sqrt{169}:\sqrt{49}=4\cdot5+13:7=20+\dfrac{13}{7}\) = \(\dfrac{153}{7}\)
c) \(\sqrt{\left(3-\sqrt{7}\right)^2}+\sqrt{7}=3-\sqrt{7}+\sqrt{7}=3\)
d) \(2\sqrt{3}-\sqrt{75}+2\sqrt{12}=2\sqrt{3}-5\sqrt{3}+4\sqrt{3}\) \(=\sqrt{3}\)
_ \(\sqrt{\dfrac{9}{4}-\sqrt{2}}=\sqrt{\left(\sqrt{2}-\sqrt{\dfrac{1}{4}}\right)^2}=\left|\sqrt{2}-\dfrac{1}{2}\right|=-\dfrac{1}{2}+\sqrt{2}=\dfrac{-1+2\sqrt{2}}{2}\)
_ \(\sqrt{\dfrac{129}{16}+\sqrt{2}}\sqrt{\left(\sqrt{8}+\sqrt{\dfrac{1}{16}}\right)^2}=\left|2\sqrt{2}+\dfrac{1}{4}\right|=2\sqrt{2}+\dfrac{1}{4}=\dfrac{1+8\sqrt{2}}{4}\)
_ \(\sqrt{\dfrac{59}{25}+\dfrac{6}{5}\sqrt{2}}=\sqrt{\left(\sqrt{2}+\sqrt{\dfrac{9}{25}}\right)^2}=\left|\sqrt{2}+\dfrac{3}{5}\right|=\sqrt{2}+\dfrac{3}{5}=\dfrac{3+5\sqrt{2}}{5}\)
_
\(\sqrt{25-4\sqrt{6}}=\sqrt{\left(2\sqrt{6}-1\right)^2}=2\sqrt{6}-1\)
\(\sqrt{16-8\sqrt{3}}=\sqrt{\left(2\sqrt{3}-2\right)^2}=2\sqrt{3}-2\)
\(\sqrt{17+12\sqrt{2}}=\sqrt{\left(2\sqrt{2}+3\right)^2}=2\sqrt{2}+3\)
\(\sqrt{21+6\sqrt{6}}=\sqrt{\left(\sqrt{3}+3\sqrt{2}\right)^2}=\sqrt{3}+3\sqrt{2}\)
\(2\sqrt{3}\). (\(3\sqrt{3}+8\sqrt{3}-5\))+ \(10\sqrt{3}\)
= \(2\sqrt{3}.3\sqrt{3}+8\sqrt{3}+10\sqrt{3}-5\)
=\(18+2\sqrt{3}-5\)
=\(13+21\sqrt{3}\)
\(\sqrt{12}+\sqrt{120}=14,41855277\)
\(\sqrt{2010}+\sqrt{2022}=89,79967785\)
\(\sqrt{25+68}+\sqrt{25+85}=163\)
\(\sqrt{25+26}+\sqrt{25}=35\)
\(\sqrt{25+66+89}=160\)
\(\sqrt{25+69+55}+\sqrt{58}+\sqrt{59}=144,2969189\)
\(\sqrt{2015+2013}=2,057888751\)
\(\sqrt{12}+\sqrt{120}=2\sqrt{30}+2\sqrt{3}=14,41855277\)
\(\sqrt{2010}+\sqrt{2022}=89,79967785\)
\(\sqrt{25+68}+\sqrt{25+85}=\sqrt{110}+\sqrt{93}=20,13173924\)
\(\sqrt{25+26}+\sqrt{25}=5+\sqrt{51}=12,14142843\)
\(\sqrt{25+66+89}=6\sqrt{5}=13,41640785\)
\(\sqrt{25+69+55}+\sqrt{58}+\sqrt{59}=27,50347447\)
\(\sqrt{258+66}=18\)
\(\sqrt{2015+2013}=63,46652661\)