# What is the square root of 190?

##### 1 Answer

It can be approximated as:

#### Explanation:

The square root of

If we factor

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

So

We can use a Newton Raphson type method to find successively better rational approximations to the irrational number

Let our first approximation be

We can use the following formula to get a better approximation:

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

where

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#