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
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