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\(\left(2^3\right)^{222}=6^{222}\)
\(\left(3^2\right)^{148}=6^{148}\)
\(\Rightarrow a>b\)
\(2^{225}=2^{3.75}=8^{75}\)
\(3^{150}=3^{2.75}=9^{75}\)
Vì 9>8 suy ra \(9^{75}>8^{75}\)Do đó: \(3^{150}>2^{225}\)
2^225=(2^15)^15=32768^15
3^150=(3^10)^15=59049^15
ta có: 32768<59049<=>32768^15<59049^15
<=>2^225<3^150
Ta có:
2225 = (23)75 = 875
3150 = (32)75 = 975
Vì 875 < 975
=> 2225 < 3150
\(2^{225}=\left(2^3\right)^{75}=8^{75}\)
\(3^{150}=\left(3^2\right)^{75}=9^{75}\)
Vì 8^75 < 9^75 nên 2^225 < 3^150
\(2^{225}=2^{3.75};3^{150}=3^{2.75}\)
\(8^{75}< 9^{75}\Rightarrow2^{225}< 3^{150}\)
\(2^{225}=\left(2^3\right)^{75}=8^{75}< 9^{75}=\left(3^2\right)^{75}=3^{150}\)
\(2^{2009}+2^{2008}+.......+2+1=b\)
\(\Rightarrow2b=2^{2010}+2^{2009}+.........+2^2+2\)
\(\Rightarrow2b-b=2^{2010}-1\Rightarrow b=2^{2010}-1\)
\(\Rightarrow A=2^{2010}-b=2^{2010}-\left(2^{2010}-1\right)=1\)
a) Ta có :
\(2^{225}=\left(2^3\right)^{75}=8^{75}\)
\(3^{150}=\left(3^2\right)^{75}=9^{75}\)
Mà 8^75 < 9^75 => 2^225<3^150
b) Ta có
2^91=(2^13)^7=8192^7
3^35=(3^5)^7=243^7
mà 8192^7<243^7=> 2^91<3^35
c) 3^4000=(3^2)^2000=9^2000
d) 2^332 < 2^333=2^3^111=8^111
3^223>3^222=9^111
=>2^332<3^223
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Ta có:\(2^{225}=\left(2^3\right)^{75}=8^{75}\)
\(3^{150}=\left(3^2\right)^{75}=9^{75}\)
\(\Rightarrow2^{225}< 3^{150}\)