What is an easy way of calculating #2^50# ?

1 Answer
Jul 31, 2017

Answer:

For an exact answer it seems that you need quite a few operations, but you can cut down the steps a bit...

Explanation:

The easiest way to explain is to start from #1# and double it #50# times. That would take while, but let's start by getting to #2^10#...

#1#
#2#
#4#
#8#
#16#
#32#
#64#
#128#
#256#
#512#
#1024#

So #2^10 = 1024#

Notice that #2^10# is only a little more than #10^3 = 1000#

Hence we find #2^50 = (2^10)^5 ~~ (10^3)^5 = 10^15#

Notice also that #1024# consists of the digits #1#, #0#, #2# and #4#, which are all relatively easy to multiply by.

Let's start by multiplying #1024xx1024#:

First note that #1024xx2 = 2048# and #2048xx2 = 4096#

So we find:

#color(white)(xx000) 1024#
#underline(xx color(white)(000) 1024)#
#color(white)(xx 000) 4096#
#color(white)(xx 00) 2048#
#underline(color(white)(xx) 1024 color(white)(000)#
#color(white)(xx) 1048576#

So #2^20 = 1048576#

Next note that #1048576xx2=2097152# and #2097152xx2=4194304#

So we find:

#color(white)(xx000) 1048576#
#underline(xx color(white)(000000)1024#
#color(white)(xx000) 4194304#
#color(white)(xx00) 2097152#
#underline(color(white)(xx) 1048576color(white)(000)#
#color(white)(xx) 1073741824#

So #2^30 = 1073741824#

Next note that #1073741824xx2 = 2147483648# and #2147483648xx2 = 4294967296#

So we find:

#color(white)(xx000) 1073741824#
#underline(xx color(white)(000000000)1024#
#color(white)(xx000)4294967296#
#color(white)(xx00)2147483648#
#underline(color(white)(xx)1073741824color(white)(000))#
#color(white)(xx)1099511627776#

So #2^40 = 1099511627776#

Next note that #1099511627776xx2=2199023255552# and #2199023255552xx2=4398046511104#

So we find:

#color(white)(xx000) 1099511627776#
#underline(xx color(white)(000000000000)1024#
#color(white)(xx000)4398046511104#
#color(white)(xx00)2199023255552#
#underline(color(white)(xx)1099511627776color(white)(000))#
#color(white)(xx)1125899906842624#

So #2^50 = 1125899906842624#

The advantages of this method are:

  • It gives an exact answer.
  • Each long multiplication only involves simple multiples #1#, #2# and #4# times.
  • Each long multiplication only involves adding #3# numbers, so the carries are not too painful.

The disadvantages are:

  • It involves doing #4# long multiplications.
  • It is still takes a fair few operations.