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July 2nd, 2014, 03:17 AM   #1
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Compare the numbers without using your calculator

A=63^63
b=129^54
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July 2nd, 2014, 04:05 AM   #2
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Please help me!!!!!!!
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July 2nd, 2014, 04:16 AM   #3
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$\displaystyle \color{blue}{\Large 63^{63}<64^{63}\ \ (1)
\\\;\\
\Large 64^{63} = 64^{9+54}=64^9\cdot64^{54}=(2^6)^9\cdot64^{54}=2^{ 54}\cdot64^{54}=(2\cdot64)^{54}=128^{54}\ \ \ (2)
\\\;\\
\Large (1), \ (2)\ \Longrightarrow\ 63^{63}<128^{54}<129^{54}\ \Longrightarrow\ 63^{63}<129^{54}}$
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July 2nd, 2014, 04:45 AM   #4
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Another method using the logarithm: (all logarithms used are base 2)

$\displaystyle 63*log(63) < 63 * log(64) = 378$

$\displaystyle 54 * log (129) > 54*log(128 ) = 378$

by consequence is

$\displaystyle 54 * log (129) > 63*log(63)$

and thus

$\displaystyle 129^{54} > 63^{63}$
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July 2nd, 2014, 05:04 AM   #5
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Guys you saved me thank you very much!!!
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