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!
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