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Formula for Finding Mth Root

Date: 08/16/2001 at 13:55:01
From: Simon
Subject: Square root or 1/m th power

I read your method of how to get square root of a number without 
using a calculator. The formula is (g + n/g)/2 or just (g^2 + n)/2g, 
where g is the guess and n is the orginal number.

So the square root of 9.5 is:

1st try, 
g = 3, n = 9.5, (3^2 + 9.5)/2(3) = 3.083333

2nd try, 
g = 3.083333, n = 9.5, (3.083333^2 + 9.5)/2(3.083333) = 3.082204

My question is: since the square root of n is the same as n^(1/2), 
what about n^(1/3) or n^(1/m) or m to the inverted mth power? Imagine 
that my calculator does not have a log function. Is there any generic 

Imagine I want 9.5^(1/3) instead of 9.5^(1/2).


Date: 08/17/2001 at 16:34:26
From: Doctor Rob
Subject: Re: Square root or 1/m th power

Thanks for writing to Ask Dr. Math, Simon.

To find an mth root, start with a guess g. Then make the next guess

    G = ((m-1)*g + n/g^(m-1))/m.

That means for cube roots, you use the formula

   G = (2*g+n/g^2)/3,

and for fifth roots, the formula

   G = (4*g+n/g^4)/5.

- Doctor Rob, The Math Forum   

Date: 06/17/2004 at 01:53:32
From: Greg
Subject: Formula for *negative* mth root

I understand your method for finding an mth root, using this 

  G = ((m-1)*g + n/g^(m-1))/m.

That is working fine for me to find mth roots.  But, it doesn't 
work to find _negative_ roots.  When m is a negative number, each
new "guess" is a wildly different number than the last "guess", 
instead of converging on the correct number.

For example, to find the -3 root of 5, and starting with an initial 
guess of 1, the formula returns these results:


The correct answer, at least according to MS Excel, is 

What formula would I use to get a negative root?

Date: 06/17/2004 at 03:17:53
From: Doctor Jacques
Subject: Re: Formula for *negative* mth root

Hi Greg,

The solution is to note that, as a^(-m) = (1/a)^m, if m is negative 
you should use the method to compute the (-m)th root of (1/n) 
instead.  To take your example, we would simply compute the 3rd root 
of 1/5 = 0.2, using the same formula (with m = + 3):

  G = (2*g + 0.2/(g^2)/3

Starting with 1, this gives the sequence of approximations:


which converges to the exact value.

In summary, to compute the (-m)th root of n, compute the mth root of 

Does this help?  Write back if you'd like to talk about this 
some more, or if you have any other questions.

- Doctor Jacques, The Math Forum

Date: 06/19/2004 at 20:39:49
From: Greg
Subject: Thank you (Formula for *negative* mth root)

Thanks, that's what I needed!
Associated Topics:
High School Exponents

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