24 Jun ##sqrt(50)## can be simplified as ##5sqrt(2)## We can also f
##sqrt(50)## can be simplified as ##5sqrt(2)## We can also find rational approximations to it. For example: ##sqrt(50) ~~ 7 14/197 ~~ 7.071066## The of ##50## is not a whole number or even a rational number. It is an irrational number but you can simplify it or find rational approximations for it. First note that ##50 = 2 xx 5 xx 5## contains a square factor ##5^2##. We can use this to simplify the square root: ##sqrt(50) = sqrt(5^2*2) = sqrt(5^2)*sqrt(2) = 5 sqrt(2)## Apart from simplifying it algebraically what is its numerical value? Note that ##7^2 = 49## so ##sqrt(49) = 7## and ##sqrt(50)## will be slightly larger than ##7##. In fact since ##50=7^2+1## the square root of ##50## is expressible as a very regular continued fraction: ##sqrt(50) = 7+1/(14+1/(14+1/(14+1/(14+1/(14+1/(14+…))))))## This can be written as ##sqrt(50) = [7;bar(14)]## where the bar over the ##14## indicates the repeating part of the continued fraction. We can terminate the continued fraction early to give us rational approximations for ##sqrt(50)##. For example: ##sqrt(50) ~~ [7;14] = 7+1/14 = 7.0bar(714285)## ##sqrt(50) ~~ [7;1414] = 7+1/(14+1/14) = 7+14/197 ~~ 7.071066## In fact: ##sqrt(50) ~~ 7.071067811865475244##
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