Associated Topics || Dr. Math Home || Search Dr. Math

### 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
formula?

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

Thanks.
Simon
```

```
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
http://mathforum.org/dr.math/
```

```
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
formula:

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:

-0.333333333333333
-0.465020576131687
-0.697963312262354
-1.32614747666764
-6.92304675837696
-3837.81145821224
-361562424747332

The correct answer, at least according to MS Excel, is
0.584803547642573.

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:

1
0.733333333
0.612855831
0.586067995
0.584806274
0.584803548

which converges to the exact value.

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

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

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search