Simplifying Square RootsDate: 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/ |
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