What is the square root of 60?

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Answer 1 · 2015-10-01T22:01:47

#sqrt(60) = 2 sqrt(15) ~~ 1921/248#

#60 = 2^2*3*5# has a square factor #2^2#

So we can simplify #sqrt(60)# using #sqrt(ab) = sqrt(a)sqrt(b)# as follows:

#sqrt(60) = sqrt(2^2 * 15) = sqrt(2^2)sqrt(15) = 2sqrt(15)#

It is not possible to simplify #sqrt(15)# further, but you can find rational approximations for it using a Newton Raphson type method.

Let #n = 15#, #p_0 = 4#, #q_0 = 1# and iterate using the formulae:

#p_(i+1) = p_i^2 + n q_i^2#

#q_(i+1) = 2 p_i q_i#

At each iteration, #p_i/q_i# is a rational approximation for #sqrt(n)#

So:

#p_1 = p_0^2 + n q_0^2 = 4^2 + 15*1^2 = 16+15 = 31#

#q_1 = 2 p_0 q_0 = 2*4*1 = 8#

Then:

#p_2 = p_1^2 + n q_1^2 = 31^2 + 15*8^2 = 961 + 960 = 1291#

#q_2 = 2 p_1 q_1 = 2 * 31 * 8 = 496#

We could go further to get a better approximation, but stop here to get:

#sqrt(15) ~~ 1291 / 496#

So

#sqrt(60) = 2sqrt(15) ~~ 2 * 1291 / 496 = 1291 / 248#