Finding a Square Root by Reasoning

Date: 11/18/2004 at 10:38:05
From: phil
Subject: what is the square root of 40,000,000

What is the square root of 40,000,000?  I don't know and it's been
bugging me all week.

Date: 11/18/2004 at 12:47:07
From: Doctor Wilko
Subject: Re: what is the square root of 40,000,000

Hi Phil,

Thanks for writing to Dr. Math!

To find the square root of 40,000,000, we need to see if there is a 
number that when multiplied by itself equals 40,000,000.  Instead of
going through the formal algebra, I'm just going to reason my way to
the answer.

Is there a "whole number" that when multiplied by itself equals 
40,000,000?  I'm just going to guess and check to see.  I know that
multiplying 2 * 2 will give me the 4, so I'll try 2,000:

  2,000 * 2,000 = 4,000,000   (too small by a factor of 10)

Now I'll try 20,000:
 
  20,000 * 20,000 = 400,000,000 (too big by a factor of 10)

So the answer we want is between those two answers.  If we could
multiply 2,000 * 2,000 * 10, we would have the correct answer!

Now remember that the square root of a number times the square root of
that same number is just that number.  For example:

  sqrt(2) * sqrt(2) = 2

Keep that in mind as you see what I do below.

     2,000 * 2,000 * sqrt(10) * sqrt(10) =

        4,000,000  *          10         =  40,000,000

So the answer to sqrt(40,000,000) is [2,000 * sqrt(10)].

Does this check?  Sure, if you multiply [2,000 * sqrt(10)] by itself, 
you'll get the correct answer.

Does this help?  Please write back if you need anything else.

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