To find the approximate weight of all the ants in the world, you can multiply the average weight of an ant by the total number of ants.
The average weight of an ant is \(3 \times 10^{-6}\) kg, and there are about \(1 \times 10^{16}\) ants.
\[ \text{Total weight} = \text{Average weight per ant} \times \text{Number of ants} \]
\[ \text{Total weight} = (3 \times 10^{-6} , \text{kg}) \times (1 \times 10^{16}) \]
To multiply these numbers in scientific notation, you multiply the coefficients and then add the exponents of the powers of 10:
\[ = 3 \times 1 \times 10^{-6 + 16} \]
\[ = 3 \times 10^{10} \]
Thus, the approximate weight of all the ants in the world is
\[ \boxed{3 \times 10^{10}} \text{ kg}. \]