How to Find Nth Roots by Hand

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Last Updated: January 19, 2020

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

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.

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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
Find the second digit of your estimate. Find a number b such that (_nC₁ a^{n - 1} (10^n-1) + _nC₂ a^{n - 2} b (10^{n - 2}) ) + . . . + _nC_{n - 1} a b^{n - 2} (10 ) + _nC_n b^{n - 1} (10⁰ ) )b is less than or equal to the modified difference above (10ⁿ (d ) + d₁d₂. . . d_n). This becomes the second digit of your estimate so far.
- The combinations notation _nC_r 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 _nC_r 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 _nP_r divided by r!
Subtract the two quantities in the last step above (10ⁿ (d ) + d₁d₂. . . d_n minus _nC₁ a^{n - 1} (10^n-1) + _nC₂ a^{n - 2} b (10^{n - 2}) ) + . . . + _nC_{n - 1} a b^{n - 2} (10 ) + _nC_n b^{n - 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 a^{n - 1} (10^{n - 1}) + _nC₂ a^{n - 2} c (10^{n - 2} ) + . . . + _nC_{n - 1} a c ^{n - 2} (10 ) + _nC_n c^{n - 1} (10⁰ ) ) c is less than or equal to the new modified difference in above (10ⁿ (d ) + d₁d₂. . . d_n). This becomes the third digit of your estimate so far.
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.
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Question

How do I find the 7th root of 4 without using a calculator?

Community Answer

You must follow the steps for the seventh root, grouping 4 as 4. 0000000 0000000, etc. And you'll likely need a 4-function calculator. "By Hand" here really means, without using the nth-root function on a scientific calculator. Though, with a lot of work, it could be done by hand.

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Question

The equation to find the second digit of your estimate as described in step 4 is too vaguely defined for me to even devise a clearly defined equation from it. Could you define it as a summation?

Keith Raskin

Community Answer

Yes, we can. As a sum: Sum 1 through k of n_C_k a^(n - k)b^(k - 1)10^(n - k).

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