Simplifying Square Roots

Date: 05/25/99 at 21:51:48
From: Susana
Subject: Simplifying Square Roots

How do I simplify a square root? What is the square root of 192?

Date: 05/26/99 at 13:35:38
From: Doctor Rick
Subject: Re: Simplifying Square Roots

Hi, Susana.

Let's start with an easy one. You can simplify sqrt(4) as 2. (This is 
how I will write square roots.) 

Too easy, right? Let's simplify sqrt(12). You can write 12 as 3 * 4 
(that's times). You can rewrite the square root of a product as the 
product of the square roots:

  sqrt(3 * 4) = sqrt(3) * sqrt(4)
              = sqrt(3) * 2

If you can factor a number into two terms, one of which is a perfect 
square, then you can simplify the square root of that number just as I 
did with 12.

How do you find the biggest perfect-square factor of a number? I do it 
by factoring the number into prime factors. Then I split it into two 
parts: one that has only even exponents, and one that has no exponent 
greater than 1. A number whose factorization has only even exponents 
is a perfect square. For instance,

  2^6 * 3^4 = (2^3)^2 * (3^2)^2
            = (2^3 * 3^2)^2

You can factor 192 into 2 * 2 * 2 * 2 * 2 * 2 * 3, or 2^6 * 3. The 
only prime factor that has an exponent greater than 1 is 2. It has an 
even exponent, so I can remove the whole thing. Therefore I factor 192 
into 64 * 3.

  sqrt(192) = sqrt(2^6 * 3)
            = sqrt(2^6) * sqrt(3)
            = 2^3 * sqrt(3)
            = 8 sqrt(3)

I hope this helps you understand how to simplify square roots.

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