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a) Có: \(4^{51}+2^{104}+4^{53}\\ =4^{51}+\left(2^2\right)^{52}+4^{53}\\ =4^{51}+4^{52}+4^{53}\\ =4^{51}\left(1+4+4^2\right)\\ =4^{51}\cdot21⋮21\left(đpcm\right)\)
b) Có: \(125^{10}+5^{31}+25^{16}\\ =\left(5^3\right)^{10}+5^{31}+\left(5^2\right)^{16}\\ =5^{30}+5^{31}+5^{32}\\ =5^{30}\left(1+5+5^2\right)\\ =5^{30}\cdot31⋮31\left(đpcm\right)\)
c) Có: \(2^{25}+4^{13}+8^9\\ =2^{25}+\left(2^2\right)^{13}+\left(2^3\right)^9\\ =2^{25}+2^{26}+2^{27}\\ =2^{23}\left(2^2+2^3+2^4\right)\\ =2^{23}\cdot28⋮28\left(đpcm\right)\)
So sánh
a)\(5^{36}\) và \(11^{24}\)
b)\(600^{^{ }25}\)và \(125^7\)
c)\(3^{^{ }21}\)và \(2^{^{ }31}\)
a)Ta có :
536=53.12=(53)12=12512
1124=112.12=(112)12=12112
Vì 125>121 nên 12512>12112
Vậy 536>1124
\(8^{30}+8^{31}+8^{32}\)
\(=8^{30}.1+8^{30}.8+8^{30}.8^2\)
\(=8^{30}.1+8^{30}.8+8^{30}.64\)
\(=8^{30}\left(1+8+64\right)\)
\(=8^{30}.73\)
\(=\left(2^3\right)^{30}.73\)
\(=2^{90}.73\)
\(=2^{89}.146⋮146\rightarrowđpcm\)
\(4^{25}+4^{26}+4^{27}+4^{28}+4^{29}+4^{30}\)
\(=4^{25}.1+4^{25}.4+4^{25}.4^2+4^{25}.4^3+4^{25}.4^4+4^{25}.4^5\)
\(=4^{25}.1+4^{25}.4+4^{25}.16+4^{25}.64+4^{25}.256+4^{25}.1024\)
\(=4^{25}\left(1+4+16+64+256+1024\right)\)
\(=4^{25}.1365\)
\(=4^{25}.195.7⋮7\rightarrowđpcm\)
\(\left(\frac{1}{2}-1\right)\left(\frac{1}{3}-1\right)\cdot\cdot\cdot\left(\frac{1}{2009}-1\right)\)
\(=\frac{-1}{2}\cdot\frac{-2}{3}\cdot\cdot\cdot\cdot\frac{-2008}{2009}\)
\(=\frac{\left(-1\right)\cdot\left(-2\right)\cdot\cdot\cdot\left(-2008\right)}{2\cdot3\cdot\cdot\cdot2009}\)
\(=\frac{1\cdot2\cdot\cdot\cdot2008}{2\cdot3\cdot\cdot\cdot2009}\)
\(=\frac{1}{2009}\)
a) Thì rất dễ
Mình làm
c) Ta có ; 2112 = (213)4 = 92614
Mà : 92614 > 544
Nên : 2112 > 544
a: \(5^{100}=\left(5^4\right)^{25}=625^{25}\)
\(8^{75}=\left(8^3\right)^{25}=512^{25}\)
mà 625>512
nên \(5^{100}>8^{75}\)
b: \(625^5=\left(5^4\right)^5=5^{20}\)
\(125^7=5^{21}\)
mà 20<21
nên \(625^5< 125^7\)
c: \(5^{23}=5^{22}\cdot5< 6\cdot5^{22}\)
d: \(7\cdot2^{13}< 8\cdot2^{13}=2^{16}\)
Ta xét :
\(5^{61}+25^{31}+125^{21}\)
\(=5^{61}+\left(5^2\right)^{31}+\left(5^3\right)^{21}\)
\(=5^{61}+5^{62}+5^{63}\)
\(=5^{61}\left(1+5+25\right)\)
\(=5^{61}.31\)
Vì \(31⋮31\)nên \(5^{61}.31⋮31\)
\(\Rightarrow5^{61}+25^{31}+125^{21}⋮31\)
\(\RightarrowĐPCM\)