Question

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

Answer 1

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.