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

Jul 31, 2017

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 ${2}^{50} = {\left({2}^{10}\right)}^{5} \approx {\left({10}^{3}\right)}^{5} = {10}^{15}$

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 $1024 \times 1024$:

First note that $1024 \times 2 = 2048$ and $2048 \times 2 = 4096$

So we find:

$\textcolor{w h i t e}{\times 000} 1024$
$\underline{\times \textcolor{w h i t e}{000} 1024}$
$\textcolor{w h i t e}{\times 000} 4096$
$\textcolor{w h i t e}{\times 00} 2048$
underline(color(white)(xx) 1024 color(white)(000)
$\textcolor{w h i t e}{\times} 1048576$

So ${2}^{20} = 1048576$

Next note that $1048576 \times 2 = 2097152$ and $2097152 \times 2 = 4194304$

So we find:

$\textcolor{w h i t e}{\times 000} 1048576$
underline(xx color(white)(000000)1024
$\textcolor{w h i t e}{\times 000} 4194304$
$\textcolor{w h i t e}{\times 00} 2097152$
underline(color(white)(xx) 1048576color(white)(000)
$\textcolor{w h i t e}{\times} 1073741824$

So ${2}^{30} = 1073741824$

Next note that $1073741824 \times 2 = 2147483648$ and $2147483648 \times 2 = 4294967296$

So we find:

$\textcolor{w h i t e}{\times 000} 1073741824$
underline(xx color(white)(000000000)1024
$\textcolor{w h i t e}{\times 000} 4294967296$
$\textcolor{w h i t e}{\times 00} 2147483648$
$\underline{\textcolor{w h i t e}{\times} 1073741824 \textcolor{w h i t e}{000}}$
$\textcolor{w h i t e}{\times} 1099511627776$

So ${2}^{40} = 1099511627776$

Next note that $1099511627776 \times 2 = 2199023255552$ and $2199023255552 \times 2 = 4398046511104$

So we find:

$\textcolor{w h i t e}{\times 000} 1099511627776$
underline(xx color(white)(000000000000)1024
$\textcolor{w h i t e}{\times 000} 4398046511104$
$\textcolor{w h i t e}{\times 00} 2199023255552$
$\underline{\textcolor{w h i t e}{\times} 1099511627776 \textcolor{w h i t e}{000}}$
$\textcolor{w h i t e}{\times} 1125899906842624$

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.

• It involves doing $4$ long multiplications.