Finding Square Roots

Date: 02/07/2002 at 22:40:11
From: Tom
Subject: Square roots

Friends at school say answers to square roots that I don't understand 
at all. I would like to know the square root of 3612 to baffle them, 
and also please teach me square roots because I'm stumped!

Date: 02/08/2002 at 11:23:26
From: Doctor Ian
Subject: Re: Square roots

Hi Tom,

If you multiply a number by itself, you get the 'square' of the 
number. The number itself is called the 'square root' of the square. 

For example, 

      2 * 2 = 2 squared = 4     <=>   2 is the square root of 4

      3 * 3 = 3 squared = 9     <=>   3 is the square root of 9

1/2 * 1/2 = 1/2 squared = 1/4   <=>   1/2 is the square root of 1/4



      x * x = x squared = x^2   <=>   x is the square root of x^2

     (-x) * (-x) = x^2          <=>   -x is the square root of x^2

This last example illustrates a tricky point: a number has _two_ 
square roots, not just one. For example, 7 is the square root of 49; 
and -7 is _also_ the square root of 49. It's easy to forget that, and 
forgetting it can get you into trouble. In most cases, however, the 
negative root is of no importance, so we ignore it. 

Anyway, to move on to your example: 60^2 is 3600, so the square root 
of 3612 has to be greater than 60; but 61^ is 3721, so the square root 
of 3612 has to be less than 61. 

So the answer is somewhere between 60 and 61.  How do you find it?  
Take a look at the Dr. Math FAQ:

   Square Roots Without a Calculator
   http://mathforum.org/dr.math/faq/faq.sqrt.by.hand.html   

If you just want to find a simpler expression, you can find the prime 
factors of the number whose root you're looking for:

  3612 = 2 * 1806

       = 2 * 2 * 903

       = 2 * 2 * 3 * 301

       = 2 * 2 * 3 * 7 * 43

Now, a rule of square roots is 

  sqrt(a*b) = sqrt(a) * sqrt(b)

so

  sqrt(3612) = sqrt(2 * 2 * 3 * 7 * 43)

             = sqrt(2 * 2) * sqrt(3 * 7 * 43)

             = 2 * sqrt(3 * 7 * 43)

             = 2 * sqrt(903)

This isn't much simpler, but sometimes this technique works very 
nicely, e.g., 

  sqrt(432) = sqrt(2 * 2 * 2 * 2 * 3 * 3 * 3)

            = sqrt(2 * 2) * sqrt(2 * 2) * sqrt(3 * 3) * sqrt(3)

            = 2 * 2 * 3 * sqrt(3)

            = 12 * sqrt(3)
   
which _is_ simpler - simple enough that there is no real need to 
convert sqrt(3) into a decimal approximation. 

Does this help? 

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

Date: 02/08/2002 at 20:48:50
From: Tom
Subject: Square roots

Thank you Docter Math. I will practice my square roots and I will 
write to you again if I need help. Letters and advice from people  
like you are worth more than anything to me. I think I might become an 
author or writer someday and I'll always think of you as my adviser 
when I was little. Also my dad explained how to do it on the 
calculator and clarified the diagrams you gave me, and now I can do 
problems like the square root of 36, which equals 6. Thank you!