## Question

###### What is the square root of 190?

## Answers

#190# has no square factors, so#sqrt(190)# does not simplify.It can be approximated as:

#11097222161/805077112 ~~ 13.784048752090222# ## Explanation:

The square root of

#190# is the non-negative number#x# such that#x^2 = 190# .If we factor

#190# then we find:

#190 = 2 * 95 = 2 * 5 * 19# So

#190# has no square factors and as a result is not possible to simplify.We can use a Newton Raphson type method to find successively better rational approximations to the irrational number

#sqrt(190)# .Let our first approximation be

#a_0 = 14# , since#14^2 = 196# is quite close.We can use the following formula to get a better approximation:

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

#n = 190# is the number for which we are trying to find the square root.See: How do you find the square root 28? for a slightly easier way of doing this. For simplicity here, I'll use the classic formula above.

Then:

#a_1 = (a_0^2 + n)/(2a_0) = (14^2+190)/(2*14) = 386/28 = 193/14 ~~ 13.7857#

#a_2 = 74489/5404 ~~ 13.78404885#

#a_3 = 11097222161/805077112 ~~ 13.784048752090222#