How Many Digits Are in the Root?

Date: 07/03/2008 at 04:21:21
From: Ankur
Subject: How to find number of digits in a square root and cube root

What is the number of digits in the cube root of a 29 digit number?

What is the method or some general formula to find this?  Also some 
formula which calculates number of digits in square root, fourth root 
etc.

In a square root the number of digits are never less than half of 
the number but can be more than that, e.g. in sq root of 13 digit
number...number of digits is 7 (this is correct because its a solution
in a book..but there was no explanation for it).

I want to know how to calculate this thing.

Date: 07/03/2008 at 19:25:49
From: Doctor Ali
Subject: Re: How to find number of digits in a square root and cube root

Hi Ankur!

Thanks for writing to Dr. Math.

We can do it this way:

We know that 29-digit numbers are between 10^28 and 10^29-1.  In 
other words, if N is a 29-digit number, we have,

  10^28  <=  N  <  10^29

Does that make sense?  Did you get why I didn't put '=' for 10^29?

Because we have all positive numbers in this inequality,  we can now 
take the square root of the sides of this inequality and have,

  Sqrt( 10^28 )  <=  Sqrt(N)  <  Sqrt( 10^29 )

We know that the number of digits in X can be evaluated using the 
following formula.

  Floor ( log_10( X ) ) + 1

Please let me know if you are not familiar with this formula or with
the floor function.

So, let's build this expression in the middle of the inequality.  
First, take the logarithm of the sides.

  Log_10(Sqrt(10^28)) <= Log_10(Sqrt(N)) < Log_10(Sqrt(10^29))

Using the rules of logarithms, you can see that,

  14 <= Log_10(Sqrt(N)) < 14.5

Now, we know that Log_10(Sqrt(N)) is between 14 and 14.5.  So, its 
floor will be 14.  Adding one, we can deduce that the square roots of
all 29-digit numbers are 15 digits long!

Can you do the same process for cube roots?

Please write back if you still have any difficulties.

- Doctor Ali, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/04/2008 at 12:16:44
From: Ankur
Subject: Thank you (How to find number of digits in a square root and
cube root)

Yes I can calculate after this....thanks a lot!!!