How do you find the square root of 36864?

`" " " "1'9'2`
`" "sqrt( 3'68'64)`
`" " " "underline(1" " " " ")`
`" " " " 268`
`" " " " underline(261" " ")`
`" " " " " "764`
`" " " " " "underline(764)`
`" " " " " " " "0`

This is how to do it.

From the decimal point of the number 36864. we group the digits by 2....so that it appears 3'68'64.
The first digit is 3. We choose a digit from 0 to 9. The best digit is 1 because 1 times 1 equals 1. If we choose 2, then 2 times 2 equals 4 which is greater than 3. So 1 is the correct digit

`" " " "1 larr`the chosen digit
`" "sqrt( 3'68'64)`
`" " " "underline(1" " " " ")`
`" " " " 268 larr`perform subtraction between 3 and 1 then bring `" " " " " " " " " " "down the next 2 digits that is 68"`

At this point we multiply our first digit 1 by 20 which is equal to 20. Think now of a digit which will be added to the result 20 and multiplied by it.

If we choose 9 then we have 20+9=29 and 29 times 9 equals 261 which is just correct because we need a number equal or lower than 268.

If we choose 8 then we have 20+8=28 and 28 times 8 equals 224 which is very much lower than 268. We need something lower but very close to 268.

Therefore 9 is the correct next digit is 9.

`" " " "1'9 larr`the next chosen digit
`" "sqrt( 3'68'64)`
`" " " "underline(1" " " " ")`
`" " " " 268 larr`bring down the next 2 digits that is 68
`" " " " underline(261" " ")`
`" " " " " "764 larr`perform subtraction to get 7 then bring down 64

At this point we multiply our previous answers 19 by 20 which is equal to 380. Think now of a digit which will be added to the result 380 and multiplied by it.

If we choose now 2 then we have 380+2=382 and 382 times 2 equals 764 which is just correct because it is equal or lower than 764.

`" " " "1'9'2 larr`the last chosen digit is 2
`" "sqrt( 3'68'64)`
`" " " "underline(1" " " " ")`
`" " " " 268 larr`bring down the next 2 digits that is 68
`" " " " underline(261" " ")`
`" " " " " "764 larr`perform subtraction to get 7 then bring down 64
`" " " " " "underline(764)`
`" " " " " " " "0 larr`the final subtraction results which ends the process.

Checking: 192 times 192 equals 36864

God bless....I hope the explanation is useful.