How do you calculate the square root of 27.3529 without using a calculator?

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Answer 1 · 2015-07-15T09:41:25

Find #sqrt(273529) = 523# then divide by #100# to get:

#sqrt(27.3529) = 5.23#

#sqrt(270000) = sqrt(3 * 3^2 * 100^2) = 300*sqrt(3)#

Hopefully you know that #sqrt(3) ~= 1.732#, so

#sqrt(270000) = 300*sqrt(3) ~= 300*1.732 = 519.6#

Let's use #520# as our first approximation for #sqrt(273529)#

Let #a_0 = 520#, #n = 273529#

Iterate using Newton Raphson method...

#a_(i+1) = a_i + (n - a_i^2)/(2a_i)#

#a_1 = a_0 + (n - a_0^2)/(2a_0)#

#=520 + (273529 - 520^2)/(2*520)#

#=520 + (273529 - 270400)/1040#

#=520 + 3129/1040#

#~=520 + 3.00865#

#~=523#

Hmmm, that looks suspicious. Try #523*523 = 273529# - yes.