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

##### 1 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#

#2#

#4#

#8#

#16#

#32#

#64#

#128#

#256#

#512#

#1024#

So

Notice that

Hence we find

Notice also that

Let's start by multiplying

First note that

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

Next note that

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

Next note that

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

Next note that

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

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.