Question

How do you calculate `5^25` by hand?

Answer 1

`5^25 = 298023223876953125`

Explanation:

`5^25 = overbrace(5xx5xx...xx5)^"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 = (10/2)^25 = 10^25/2^25`

Let's go...

`10000000000000000000000000`

`color(white)(0)5000000000000000000000000`
`color(white)(0)2500000000000000000000000`
`color(white)(0)1250000000000000000000000`
`color(white)(00)625000000000000000000000`
`color(white)(00)312500000000000000000000`

`color(white)(00)156250000000000000000000`
`color(white)(000)78125000000000000000000`
`color(white)(000)39062500000000000000000`
`color(white)(000)19531250000000000000000`
`color(white)(0000)9765625000000000000000`

`color(white)(0000)4882812500000000000000`
`color(white)(0000)2441406250000000000000`
`color(white)(0000)1220703125000000000000`
`color(white)(00000)610351562500000000000`
`color(white)(00000)305175781250000000000`

`color(white)(00000)152587890625000000000`
`color(white)(000000)76293945312500000000`
`color(white)(000000)38146972656250000000`
`color(white)(000000)19073486328125000000`
`color(white)(0000000)9536743164062500000`

`color(white)(0000000)4768371582031250000`
`color(white)(0000000)2384185791015625000`
`color(white)(0000000)1192092895507812500`
`color(white)(00000000)596046447753906250`
`color(white)(00000000)298023223876953125`