How to Find Nth Roots by Hand

Step-by-Step Guide

1. 1

   Step 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. , 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., 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.), 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! , 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.), 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., 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.
2. 2

   Step 2: Find an initial estimate.

3. 3

   Step 3: Modify the difference.

4. 4

   Step 4: Find the second digit of your estimate.

5. 5

   Step 5: Find your new modified difference.

6. 6

   Step 6: Find the third digit of your estimate.

7. 7

   Step 7: Repeat.

Detailed Guide

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