Square root of 18769

How do I work out the Square root of 18769?

Answers
Firstly, try to express 18769 as a product of prime factors to make numbers easier to work with. Secondly, take the square root of the simplified prime factors expression and there should be your answer.
mathstutor80
31 October 2014
Calculating square roots by hand has rather gone out of fashion since the invention of log tables, let alone hand calculators, but I seem to remember my old dad demonstrating a method.  This one depends on an iterative approach that gradually reduces an error until it's small enough not to matter.  Imagine a square.  You know that its area is 18769 but you don't know its side.  You can estimate it to the nearest hundred though.  13^2 is 169 and 14^2 is 196, so you know that your root has to be between 130 and 140.  Now imagine your original square with a 130 x 130 square chopped out of it.  How much area is left?  1869.  Now scrape away this area.  If you added another 5 units that's 2x130x5 = 1300, leaving 569.  Now you know that your root is close to 135.  Go round the loop again,  135^2 = 18225.  You now have 544 area units left.  544/135 = approx 4   (in other words, 2x135x2 = 540).  So, the root is close to 137.  Sure enough, 137^2=18769.
ianmoth
02 November 2014
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