# What is the square root of 190?

Sep 16, 2015

$190$ has no square factors, so $\sqrt{190}$ does not simplify.

It can be approximated as:

$\frac{11097222161}{805077112} \approx 13.784048752090222$

#### Explanation:

The square root of $190$ is the non-negative number $x$ such that ${x}^{2} = 190$.

If we factor $190$ then we find:

$190 = 2 \cdot 95 = 2 \cdot 5 \cdot 19$

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:

${a}_{i + 1} = \frac{{a}_{i}^{2} + n}{2 {a}_{i}}$

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:

${a}_{1} = \frac{{a}_{0}^{2} + n}{2 {a}_{0}} = \frac{{14}^{2} + 190}{2 \cdot 14} = \frac{386}{28} = \frac{193}{14} \approx 13.7857$

${a}_{2} = \frac{74489}{5404} \approx 13.78404885$

${a}_{3} = \frac{11097222161}{805077112} \approx 13.784048752090222$