Elements of a square root and calculation step by step

We group the digits two by two and $47.19.69$ is obtained in the radicand. We look for a number that, when squared, gives $47$. The closest number is $6$ because $6\cdot6 = 36$.

This result is subtracted from $47$ and the following digits $(19)$ are moved down. We also separate the last number.

Then write the double of $6$, which is $12$. We divide $111$ by $12$, to obtain the number that it is necessary to add and to multiply. In this case it is $8$.

It is subtracted and the following group of two digits are moved down and the last digit is separated.

We repeat the previous step.

The double of $68$ is written below, $136$. Divide $956$ by $136$, to obtain the number that it is necessary to add and to multiply. In this case it is $7$, which must be added to the first line.

Operating the remainder is zero, therefore, the first line is taken and that number is the root.