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\(a,16^{19}=\left(2^4\right)^{19}=2^{76}\\ 8^{25}=\left(2^3\right)^{25}=2^{75}\)
Vì \(2^{76}>2^{75}=>16^{19}>8^{25}\)
b,\(3^{500}=\left(3^5\right)^{100}=243^{100}\)
Vì \(243^{100}>5^{100}=>3^{500}>5^{100}\)
\(3^{99}=\left(3^3\right)^{33}=27^{33}>27^{21}>11^{21}\\ 16^x< 128^4\\ \Rightarrow\left(2^4\right)^x< \left(2^7\right)^4\\ \Rightarrow2^{4x}< 2^{28}\Rightarrow4x< 28\Rightarrow x< 7\)
\(1,\\ 16^x< 128^4\Rightarrow\left(2^4\right)^x< \left(2^6\right)^4\Rightarrow2^{4x}< 2^{24}\\ \Rightarrow4x=24\Rightarrow x=6\\ 2,\\ 3^{99}=\left(3^3\right)^{33}=27^{33}>27^{21}>11^{21}\)
a: Ta có: \(3^{2x+1}< 27\)
\(\Leftrightarrow2x+1< 3\)
\(\Leftrightarrow x< 1\)
hay x=0
a: 99^20=9801^10<9999^10
b: 3^500=243^100
5^300=125^300
=>3^500>5^300
Giải:
a) Gọi dãy đó là A, ta có:
\(A=\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{2014}}\)
\(2A=\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{2013}}\)
\(2A-A=\left(\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{2013}}\right)-\left(\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{2014}}\right)\)
\(A=\dfrac{1}{2}-\dfrac{1}{2^{2014}}\)
Vì \(\dfrac{1}{2}< 1;\dfrac{1}{2^{2014}}< 1\) nên \(\dfrac{1}{2}-\dfrac{1}{2^{2014}}< 1\)
\(\Rightarrow A< 1\)
b) \(A=\dfrac{10^{11}-1}{10^{12}-1}\) và \(B=\dfrac{10^{10}+1}{10^{11}+1}\)
Ta có:
\(A=\dfrac{10^{11}-1}{10^{12}-1}\)
\(10A=\dfrac{10^{12}-10}{10^{12}-1}\)
\(10A=\dfrac{10^{12}-1+9}{10^{12}-1}\)
\(10A=1+\dfrac{9}{10^{12}-1}\)
Tương tự:
\(B=\dfrac{10^{10}+1}{10^{11}+1}\)
\(10B=\dfrac{10^{11}+10}{10^{11}+1}\)
\(10B=\dfrac{10^{11}+1+9}{10^{11}+1}\)
\(10B=1+\dfrac{9}{10^{11}+1}\)
Vì \(\dfrac{9}{10^{12}-1}< \dfrac{9}{10^{11}+1}\) nên \(10A< 10B\)
\(\Rightarrow A< B\)
a) \(5^{48}=\left(5^4\right)^{12}=625^{12}\)
\(2^{108}=\left(2^9\right)^{12}=512^{12}\)
Do \(625>512\Rightarrow625^{12}>512^{12}\) \(\Rightarrow5^{48}>2^{108}\) (1)
Lại có: \(108>105\Rightarrow2^{108}>2^{105}\) (2)
Từ (1) và (2) \(\Rightarrow5^{48}>2^{105}\)
b) \(2^{50}=\left(2^5\right)^{10}=32^{10}\)
Do \(33>32\Rightarrow33^{10}>32^{10}\)
Vậy \(33^{10}>2^{50}\)
c) Do \(513>512\Rightarrow513^{100}>512^{100}\) (1)
\(512^{100}=\left(2^9\right)^{100}=2^{900}\) \(=2^{10.90}=\left(2^{10}\right)^{90}=1024^{90}\) (2)
Do \(1024>1023\Rightarrow1024^{90}>1023^{90}\) (3)
Từ (1), (2) và (3) \(\Rightarrow513^{100}>1023^{90}\)
b)
a = 25.26 261 = 25.(26 260 +1) = 25.10.2626 + 25 = 25.10.26.101 + 25
b = 26.25 251 = 26.(25 250 + 1) = 26.10.2525 + 26 = 26.10.25.101 + 26
Suy ra a < b
a) 1255=(53)5=515
257=(52)7=514
Vì 15>14 nên 515>514
hay 1255>257
b) 354=32.27=(32)27=927
281=23.27=(23)27=827
Vì 9>8 nên 927>827
hay 354>281