Question 257907: Without using a calculator or computer, determine which is larger, 2^3000 or 3^2000. Found 2 solutions by richwmiller, CharlesG2:Answer by richwmiller(17219) (Show Source):
You can put this solution on YOUR website! Without using a calculator or computer, determine which is larger, 2^3000 or 3^2000
2^1=2
2^2=4
2^3=8
2^4=16
2^5=32
2^10=32*32=32*30+32*2=960+64=1024
2^9=512
3^1=3
3^2=9
3^3=27
3^4=27*3=20*3+7*3=60+21=81
3^5=243
3^6=729
3^10=243*243=243*240+243*3=243*24*10+240*3+3*3=(243*20+243*4)*10+720+9
3^10=(243*2*10+243*4)*10+720+9
3^10=(4860+972)*10+729
3^10=(4860+900+40+32)*10+729
3^10=(5760+40+32)*10+729
3^10=58320+729=59020+29=59049 (yes did this without a calculator , just a matter of breaking it down into simpler parts)
do not think will go much further in powers, it begins to get really hard without a calculator
but it seems obvious to me that 3 to a power is climbing a lot faster than 2 to a power
so I am gonna say 3^2000 is bigger
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