Question

What is the square root of 190?

Answer 1

`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`