Find the largest possible root of a number that is whole Using 8 as an example radicand, the degree would be 3 because ∜8 is not a whole number, while √8 is not the largest possible whole root. This type of problem is easy to calculate mentally with small numbers, but for large numbers it gets tricky. Is there a way to calculate it without iterating through degrees until the right one is found? It seems the answers may involve logarithms.

Braeden Valenzuela

Braeden Valenzuela

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2022-08-21

Find the largest possible root of a number that is whole
Using 8 as an example radicand, the degree would be 3 because ∜8 is not a whole number, while √8 is not the largest possible whole root. This type of problem is easy to calculate mentally with small numbers, but for large numbers it gets tricky. Is there a way to calculate it without iterating through degrees until the right one is found? It seems the answers may involve logarithms.
Thanks in advance.

Answer & Explanation

Willow Avery

Willow Avery

Beginner2022-08-22Added 11 answers

If you can find the prime factorization of the number, take the greatest common divisor of all the exponents in it.
Unfortunately factoring large numbers is not quick, so simply checking all possible degrees up to log 2 of the number might well be faster asymptotically.
For most inputs, a combination might be the best strategy -- look for small prime factors, and take the gcd of their exponents. Then you only need to check degrees that are factors of that gcd.
alan3885uo

alan3885uo

Beginner2022-08-23Added 3 answers

If a has prime factorization
a = k = 1 n p k e k  where  e k N , p k  prime
then
a n N  exactly if  n e k  for all  k , meaning if  n gcd ( e 1 , , d k ) .
The largest such n is thus gcd ( e 1 , , d k )

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