How do you calculate 5^25 by hand?

Jan 21, 2017

${5}^{25} = 298023223876953125$

Explanation:

${5}^{25} = {\overbrace{5 \times 5 \times \ldots \times 5}}^{\text{25 times}}$

To calculate it manually, it may be easier to start with ${10}^{25}$ and divide it by $2$, $25$ times, since:

${5}^{25} = {\left(\frac{10}{2}\right)}^{25} = {10}^{25} / {2}^{25}$

Let's go...

$10000000000000000000000000$

$\textcolor{w h i t e}{0} 5000000000000000000000000$
$\textcolor{w h i t e}{0} 2500000000000000000000000$
$\textcolor{w h i t e}{0} 1250000000000000000000000$
$\textcolor{w h i t e}{00} 625000000000000000000000$
$\textcolor{w h i t e}{00} 312500000000000000000000$

$\textcolor{w h i t e}{00} 156250000000000000000000$
$\textcolor{w h i t e}{000} 78125000000000000000000$
$\textcolor{w h i t e}{000} 39062500000000000000000$
$\textcolor{w h i t e}{000} 19531250000000000000000$
$\textcolor{w h i t e}{0000} 9765625000000000000000$

$\textcolor{w h i t e}{0000} 4882812500000000000000$
$\textcolor{w h i t e}{0000} 2441406250000000000000$
$\textcolor{w h i t e}{0000} 1220703125000000000000$
$\textcolor{w h i t e}{00000} 610351562500000000000$
$\textcolor{w h i t e}{00000} 305175781250000000000$

$\textcolor{w h i t e}{00000} 152587890625000000000$
$\textcolor{w h i t e}{000000} 76293945312500000000$
$\textcolor{w h i t e}{000000} 38146972656250000000$
$\textcolor{w h i t e}{000000} 19073486328125000000$
$\textcolor{w h i t e}{0000000} 9536743164062500000$

$\textcolor{w h i t e}{0000000} 4768371582031250000$
$\textcolor{w h i t e}{0000000} 2384185791015625000$
$\textcolor{w h i t e}{0000000} 1192092895507812500$
$\textcolor{w h i t e}{00000000} 596046447753906250$
$\textcolor{w h i t e}{00000000} 298023223876953125$