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Xét \(\frac{1}{\sqrt{13}}>\frac{1}{\sqrt{14}}\Rightarrow\frac{1}{\sqrt{13}}-1< \frac{1}{\sqrt{14}}+1\)
Mà \(\sqrt{225}< \sqrt{289}\)
\(\Rightarrow\sqrt{225}-\left(\frac{1}{\sqrt{13}}-1\right)< \sqrt{289}-\left(\frac{1}{\sqrt{14}}+1\right)\)
Vậy....................
a: \(\left(18\dfrac{1}{3}:\sqrt{225}+8\dfrac{2}{3}\cdot\sqrt{\dfrac{49}{4}}\right):\left[\left(12\dfrac{1}{3}+8\dfrac{6}{7}\right)-\dfrac{\left(\sqrt{7}\right)^2}{\left(3\sqrt{2}\right)^2}\right]:\dfrac{1704}{445}\)
\(=\left(\dfrac{55}{3}:15+\dfrac{26}{3}\cdot\dfrac{7}{4}\right):\left[\left(12+\dfrac{1}{3}+8+\dfrac{6}{7}\right)-\dfrac{7}{18}\right]\cdot\dfrac{445}{1704}\)
\(=\left(\dfrac{55}{45}+\dfrac{91}{6}\right):\left[20+\dfrac{101}{126}\right]\cdot\dfrac{445}{1704}\)
\(=\dfrac{295}{18}:\dfrac{2621}{126}\cdot\dfrac{445}{1704}\)
\(=\dfrac{295}{18}\cdot\dfrac{126}{2621}\cdot\dfrac{445}{1704}\simeq0,21\)
b: \(\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+...+\dfrac{1}{99\cdot100}\)
\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{99}-\dfrac{1}{100}\)
\(=1-\dfrac{1}{100}=\dfrac{99}{100}\)
c: \(\left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)\cdot...\cdot\left(1-\dfrac{1}{n+1}\right)\)
\(=\dfrac{1}{2}\cdot\dfrac{2}{3}\cdot...\cdot\dfrac{n}{n+1}\)
\(=\dfrac{1}{n+1}\)
d: \(-66\left(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{11}\right)+124\cdot\left(-37\right)+63\cdot\left(-124\right)\)
\(=-66\cdot\dfrac{33-22+6}{66}+124\left(-37-63\right)\)
\(=-17-12400=-12417\)
e: \(\dfrac{7}{4}\left(\dfrac{33}{12}+\dfrac{3333}{2020}+\dfrac{333333}{303030}+\dfrac{33333333}{42424242}\right)\)
\(=\dfrac{7}{4}\left(\dfrac{33}{12}+\dfrac{33}{20}+\dfrac{33}{30}+\dfrac{33}{42}\right)\)
\(=\dfrac{7}{4}\cdot33\cdot\left(\dfrac{1}{12}+\dfrac{1}{20}+\dfrac{1}{30}+\dfrac{1}{42}\right)\)
\(=33\cdot\dfrac{7}{4}\left(\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}\right)\)
\(=33\cdot\dfrac{7}{4}\cdot\left(\dfrac{1}{3}-\dfrac{1}{7}\right)\)
\(=33\cdot\dfrac{7}{4}\cdot\dfrac{4}{21}=\dfrac{33\cdot1}{3}=11\)
M=\(\left(\dfrac{55}{3}:15+\dfrac{26}{3}.\dfrac{7}{2}\right):\left[\left(\dfrac{37}{3}+\dfrac{62}{7}\right)-\dfrac{7}{18}\right]:\dfrac{1704}{445}\)
M=\(\left(\dfrac{11}{9}+\dfrac{91}{3}\right):\left[\dfrac{445}{21}-\dfrac{7}{18}\right]:\dfrac{1704}{445}\)
M=\(\dfrac{284}{9}:\dfrac{2621}{126}:\dfrac{1704}{445}\)
M=\(\dfrac{3115}{7863}\)
\(x=\left(1-\dfrac{1}{\sqrt{4}}\right).\left(1-\dfrac{1}{\sqrt{16}}\right).\left(1-\dfrac{1}{\sqrt{36}}\right).\left(1-\dfrac{1}{\sqrt{64}}\right).\left(1-\dfrac{1}{\sqrt{100}}\right)\)
\(x=\left(1-\dfrac{1}{2}\right).\left(1-\dfrac{1}{4}\right).\left(1-\dfrac{1}{6}\right).\left(1-\dfrac{1}{8}\right).\left(1-\dfrac{1}{10}\right)\)
\(x=\dfrac{1}{2}.\dfrac{3}{4}.\dfrac{5}{6}.\dfrac{7}{8}.\dfrac{9}{10}\)
\(x=\dfrac{63}{256}\)
và \(y=\sqrt{20+0,25}\)
\(y=\sqrt{20,25}\)
\(y=4,5\)
Do 4,5 > \(\dfrac{63}{256}\)
=> x<y
\(B=\dfrac{\left(\dfrac{5}{70}-\dfrac{10\sqrt{2}}{70}+\dfrac{6\sqrt{2}}{70}\right)\cdot\dfrac{-4}{15}}{\left(\dfrac{5}{50}+\dfrac{6\sqrt{2}}{50}-\dfrac{10\sqrt{2}}{50}\right)\cdot\dfrac{5}{7}}=\dfrac{\dfrac{5-4\sqrt{2}}{70}\cdot\dfrac{-4}{15}}{\dfrac{5-4\sqrt{2}}{50}\cdot\dfrac{5}{7}}\)
\(=\dfrac{-4\left(5-4\sqrt{2}\right)}{70\cdot15}\cdot\dfrac{50\cdot7}{5\left(5-4\sqrt{2}\right)}=\dfrac{-4}{5}\cdot\dfrac{350}{70\cdot15}=\dfrac{-4}{5}\cdot\dfrac{1}{3}=\dfrac{-4}{15}\)
Ta có :
\(\sqrt{225}-\left(\dfrac{1}{\sqrt{13}}-1\right)=15-\dfrac{1}{\sqrt{13}}+1=16-\dfrac{1}{\sqrt{13}}\)
\(\sqrt{289}-\left(\dfrac{1}{\sqrt{14}}+1\right)=17-\dfrac{1}{\sqrt{14}}-1=16-\dfrac{1}{\sqrt{14}}\)
Vì 13 < 14 \(\Rightarrow\sqrt{13}< \sqrt{14}\)
\(\Rightarrow\dfrac{1}{\sqrt{13}}>\dfrac{1}{\sqrt{14}}\)
\(\Rightarrow16-\dfrac{1}{\sqrt{13}}< 16-\dfrac{1}{\sqrt{14}}\)
\(\Rightarrow\sqrt{225}-\left(\dfrac{1}{\sqrt{13}}-1\right)< \sqrt{289}-\left(\dfrac{1}{\sqrt{14}}+1\right)\)
Ta có: \(\sqrt{225}-\left(\dfrac{1}{\sqrt{13}}-1\right)\)
\(=15-\dfrac{1}{\sqrt{13}}+1\)
\(=\left(15+1\right)-\dfrac{1}{\sqrt{13}}\)
\(=16-\dfrac{1}{\sqrt{13}}\)
Và: \(\sqrt{289}-\left(\dfrac{1}{\sqrt{14}}+1\right)\)
\(=17-\dfrac{1}{\sqrt{14}}-1\)
\(=\left(17-1\right)-\dfrac{1}{\sqrt{14}}\)
\(=16-\dfrac{1}{\sqrt{14}}\)
Vì \(13< 14\Rightarrow\sqrt{13}< \sqrt{14}\Rightarrow\dfrac{1}{\sqrt{13}}>\dfrac{1}{\sqrt{14}}\Rightarrow-\dfrac{1}{\sqrt{13}}< -\dfrac{1}{\sqrt{14}}\Rightarrow16-\dfrac{1}{\sqrt{13}}< 16-\dfrac{1}{\sqrt{14}}\)
Hay \(\sqrt{225}-\left(\dfrac{1}{\sqrt{13}}-1\right)< \sqrt{289}-\left(\dfrac{1}{\sqrt{14}}+1\right)\)
Chúc bn học tốt
