Adding Square Roots

Date: 12/26/2001 at 14:22:26
From: Ollie
Subject: Adding square roots

I'm going over a practice test for an intermediate algebra assessment 
test. One of the questions is this:

?3+?27 = ?
(or, in English)
square root of 3 + square root of 27 = ?

I thought the answer was 3+?3 (3 + square root 3). However, there is 
an answer key and the answer is actually 4?3 (4 square root 3). I 
don't understand where I went wrong. Can you help?

-Ollie

Date: 12/26/2001 at 15:27:44
From: Doctor Achilles
Subject: Re: Adding square roots

Hi Ollie,

Thanks for writing to Dr. Math.

First, just for terminology, I prefer this for the square root:

  sqrt(x)

So you have:

  sqrt(3) + sqrt(27) = ?

What I do whenever I'm presented with roots (square roots, cube roots, 
fourth roots, whatever) is to factor the number inside down to primes.  
Remember prime factorization? It turns out it is actually useful for 
this.

First, let's factor 3 to primes:  well, that was easy, 3 _is_ prime.

Next, let's factor 27 to primes:  27 = 9 * 3 = 3 * 3 * 3.

Ok, now we have:

  sqrt(3) + sqrt(3*3*3) = ?

The way to simplify roots (once you've factored the insides down to 
prime numbers) is to look for factors that appear more than once. The 
rule is: 

   for a square root, you can remove two identical factors 
   from inside the sqrt symbol and bring one copy of them outside 

  (for cube roots, you need 3 identical factors inside to bring 
   one out, for fourth roots you need 4, etc.)

So is there any factor that appears twice?  Yes, there are two 3's.  
So let's get rid of two 3's inside the square root, and put one 
outside (multiplied by the square root), like this:

  sqrt(3) + 3*sqrt(3) = ?

Now there are no more duplicate factors inside either square root, so 
we're done simplifying the roots. Can we do anything else?

Sure, we have one sqrt(3) plus three sqrt(3)'s, so we can add them 
together and end up with:

  4*sqrt(3)

I hope this helps. If you have other questions about this or you're 
still stuck, please write back.

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

Date: 12/26/2001 at 22:10:57
From: Ollie
Subject: Adding square roots

Thank you very much!  That did explain it very well. I knew there had 
to be a "trick" to finding the answer easier than just doing the raw 
math. You're my new hero.  :)

-Ollie