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July 2nd, 2014, 03:17 AM  #1 
Newbie Joined: Jun 2014 From: Ioannina Posts: 8 Thanks: 0  Compare the numbers without using your calculator
A=63^63 b=129^54 
July 2nd, 2014, 04:05 AM  #2 
Newbie Joined: Jun 2014 From: Ioannina Posts: 8 Thanks: 0 
Please help me!!!!!!!

July 2nd, 2014, 04:16 AM  #3 
Senior Member Joined: Apr 2014 From: Europa Posts: 584 Thanks: 177 
$\displaystyle \color{blue}{\Large 63^{63}<64^{63}\ \ (1) \\\;\\ \Large 64^{63} = 64^{9+54}=64^9\cdot64^{54}=(2^6)^9\cdot64^{54}=2^{ 54}\cdot64^{54}=(2\cdot64)^{54}=128^{54}\ \ \ (2) \\\;\\ \Large (1), \ (2)\ \Longrightarrow\ 63^{63}<128^{54}<129^{54}\ \Longrightarrow\ 63^{63}<129^{54}}$ 
July 2nd, 2014, 04:45 AM  #4 
Senior Member Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11 
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 $\displaystyle 129^{54} > 63^{63}$ 
July 2nd, 2014, 05:04 AM  #5 
Newbie Joined: Jun 2014 From: Ioannina Posts: 8 Thanks: 0 
Guys you saved me thank you very much!!!


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