# How to Find Nth Roots by Hand

Here is that funny long division-like method for finding square and cube roots generalized to nth roots. These are all really extensions of the Binomial Theorem.

## Steps

1. 1
Partition your number. Separate the number you want to find the nth root of into n-digit intervals before and after the decimal. If there are fewer than n digits before the decimal, then that is the first interval. And if there are no digits or fewer than n digits after the decimal, fill in the spaces with zeroes.
2. 2
Find an initial estimate. Find a number (a) raised to the nth power closest to the first n digits (or the fewer than n digits before the decimal) as a base-ten number without going over. This is the first and only digit of your estimate so far.
3. 3
Modify the difference. Subtract your estimate to the nth power (an) from those first n digits and bring down the next n digits next to that difference to form a new number, a modified difference. (Or multiply the difference by 10n and add the next n digits as a base-ten number.)
4. 4
Find the second digit of your estimate. Find a number b such that (nC1 an - 1 (10n-1) + nC2 an - 2 b (10n - 2) ) + . . . + nCn - 1 a bn - 2 (10 ) + nCn bn - 1 (100 ) )b is less than or equal to the modified difference above (10n (d ) + d1d2. . . dn). This becomes the second digit of your estimate so far.
• The combinations notation nCr represents n! divided by the product of (n - r)! and r!, where n! = n(n - 1)(n - 2)(n - 3) . . . (3)(2)(1). The notation nCr is sometimes expressed as n over r within tall parentheses without a division bar, and it can be calculated simply as the first r factors of n! divided by r!, which is often written as nPr divided by r!
5. 5
Find your new modified difference. Subtract the two quantities in the last step above (10n (d ) + d1d2. . . dn minus nC1 an - 1 (10n-1) + nC2 an - 2 b (10n - 2) ) + . . . + nCn - 1 a bn - 2 (10 ) + nCn bn - 1 (100 ) )b) to form your new modified difference by bringing down the next set of n digits next to that result. (Or multiply the difference by 10n and add the next n digits as a base-ten number.)
6. 6
Find the third digit of your estimate. Find a new number c and use your estimate so far, a (which is now 2 digits), such that (nC1 an - 1 (10n - 1) + nC2 an - 2 c (10n - 2 ) + . . . + nCn - 1 a c n - 2 (10 ) + nCn cn - 1 (100 ) ) c is less than or equal to the new modified difference in above (10n (d ) + d1d2. . . dn). This becomes the third digit of your estimate so far.
7. 7
Repeat. Keep repeating the last two steps above to find more digits of your estimate.
• This is basically a rolling binomial expansion minus the lead term, where the two terms involved are the prior estimate multiplied by 10 and the next digit to improve the estimate.

## Article Info

Categories: Algebra