Why does the method to find out log and cube roots work? To find cube roots of any number with a si

Adriana Ayers 2022-06-26 Answered
Why does the method to find out log and cube roots work?
To find cube roots of any number with a simple calculator, the following method was given to us by our teacher, which is accurate to atleast one-tenths.
1)Take the number X, whose cube root needs to be found out, and take its square root 13 times (or 10 times) i.e. . . . . X
2)next, subtract 1, divide by 3 (for cube root, and any number n for nth root), add 1.
3) Then square the resultant number (say c) 13times (or 10 times if you had taken out root 10 times) i.e. c 2 2 . . . .2 = c 2 13 . This yields the answer.
I am not sure whether taking the square root and the squares is limited to 10/13 times, but what I know is this method does yield answers accurate to atleast one-tenths.
For finding the log, the method is similar:-
1)Take 13 times square root of the number, subtract 1, and multiply by 3558. This yield s the answer.
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Answers (1)

Paxton James
Answered 2022-06-27 Author has 25 answers
Let's use these classical formulae :
e x = lim n ( 1 + x n ) n
ln x = lim n n ( x 1 / n 1 )
to get (replacing the limit by a large enough value of n: N = 2 13 ) :
x 3 = e ( ln x / 3 ) ( 1 + ln x / 3 N ) N ( 1 + N ( x 1 / N 1 ) 3 N ) N ( 1 + ( x 1 / N 1 ) 3 ) N
Concerning the decimal logarithm we have :
log 10 x = ln x ln 10 N ln 10 ( x 1 / N 1 )
For N = 2 13 we may (as indicated by peterwhy) approximate the fraction with
N ln 10 = 2 13 ln 10 0.4343 × 8192 3558
Hoping this clarified things,
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