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a) \(\left(\frac{1}{243}\right)^9=\left(\frac{1}{3^5}\right)^9=\frac{1}{3^{45}}\)
\(\left(\frac{1}{83}\right)^{13}< \left(\frac{1}{81}\right)^{13}=\left(\frac{1}{3^4}\right)^{13}=\frac{1}{3^{52}}< \frac{1}{3^{45}}=\left(\frac{1}{243}\right)^9\Rightarrow\left(\frac{1}{83}\right)^{13}< \left(\frac{1}{243}\right)^9\)
b) 199010 + 19909
= 19909 ( 1990 + 1 )
= 19909 . 1991 < 199110 = 19919 . 1991
Vậy 199010 + 19909 < 199110
ta co( \(\frac{1}{243}\))9=(\(\frac{1}{3}\))45=(\(\frac{1}{81}\))11,25<(\(\frac{1}{83}\))13
ta co( \(\frac{1}{243}\))9=(\(\frac{1}{3}\))45=(\(\frac{1}{81}\))11,25<(\(\frac{1}{83}\))13
a: 51/56=1-5/56
61/66=1-5/66
mà -5/56<-5/66
nên 51/56<61/66
b: 41/43<1<172/165
c: \(\dfrac{101}{506}>0>-\dfrac{707}{3534}\)
Ta có: \(S< \dfrac{1}{2}+\dfrac{1}{11}+\dfrac{1}{11}+\dfrac{1}{11}+\dfrac{1}{31}+\dfrac{1}{31}+\dfrac{1}{32}+\dfrac{1}{32}\) \(=\dfrac{1}{2}+\dfrac{3}{11}+\dfrac{2}{31}+\dfrac{2}{32}\)
\(=\dfrac{4909}{5456}< \dfrac{9}{10}\)
\(\Rightarrow S< \dfrac{9}{10}\)
Vậy \(S< \dfrac{9}{10}\)
1 )Ta có
\(M=\left(\dfrac{1}{2^2}-1\right)\cdot\left(\dfrac{1}{3^2}-1\right)\cdot\left(\dfrac{1}{4^2}-1\right)...\left(\dfrac{1}{100^2}-1\right)\)
\(=\left(\dfrac{1}{2}-1\right)\left(\dfrac{1}{2}+1\right)\left(\dfrac{1}{3}-1\right)\left(\dfrac{1}{3}+1\right).....\left(\dfrac{1}{100}-1\right)\left(\dfrac{1}{100}+1\right)\)
\(=\dfrac{-1}{2}\cdot\dfrac{3}{2}\cdot\dfrac{-2}{3}\cdot\dfrac{4}{3}\cdot\dfrac{-3}{4}\cdot\dfrac{5}{4}\cdot\cdot\cdot\cdot\dfrac{-99}{100}\cdot\dfrac{101}{100}\)
\(=\dfrac{-1\cdot\left(-2\right)\cdot\left(-3\right)\cdot3\cdot\left(-4\right)\cdot4\cdot\left(-5\right)\cdot5....\cdot\left(-100\right)\cdot100\cdot101}{2^2\cdot3^2\cdot4^2....\cdot100^2}\)
\(=-\dfrac{101}{200}< \dfrac{1}{2}\)
2 ) Số phân số của biểu thức B là 180 phân số
Ta có
\(\dfrac{1}{20}>\dfrac{1}{200};\dfrac{1}{21}>\dfrac{1}{200};\dfrac{1}{22}>\dfrac{1}{200};....;\dfrac{1}{199}>\dfrac{1}{200}\)
\(\Rightarrow B=\dfrac{1}{20}+\dfrac{1}{21}+...+\dfrac{1}{200}>\dfrac{1}{200}\cdot180=\dfrac{9}{10}\)
Sửa đề: \(\left(\dfrac{1}{81}\right)^{13}\)
Ta có: \(\left(\dfrac{1}{243}\right)^9=\left(\dfrac{1}{3}\right)^{45}\)
\(\left(\dfrac{1}{81}\right)^{13}=\left(\dfrac{1}{3}\right)^{52}\)
mà \(\left(\dfrac{1}{3}\right)^{45}< \left(\dfrac{1}{3}\right)^{52}\)
nên \(\left(\dfrac{1}{243}\right)^9< \left(\dfrac{1}{81}\right)^{13}\)
Uce! Tks!