What is the square root of 190?

#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#