Answer
At $~~x = 4.00~cm~~$ the net magnetic field is zero.
Work Step by Step
If the net magnetic field is zero, the magnetic field due to each current must be equal in magnitude and opposite in direction. By the right hand rule, this point must be somewhere between the two currents.
Let $x$ be the point where the net magnetic field is zero. Then the distance from $i_2$ is $~~(16.0~cm-x)$
We can write the general expression for the magnetic field produced by a current in a straight wire:
$B = \frac{\mu_0~i}{2~\pi~R}$
To find $x$, we can equate the magnitude of the magnetic field due to each current:
$\frac{\mu_0~i_1}{2~\pi~x} = \frac{\mu_0~i_2}{(2~\pi)~(0.160-x)}$
$\frac{i_1}{x} = \frac{3.00~i_1}{0.160-x}$
$\frac{1}{x} = \frac{3.00}{0.160-x}$
$0.160-x = 3.00~x$
$4.00~x = 0.160$
$x = 0.0400~m$
$x = 4.00~cm$
At $~~x = 4.00~cm~~$ the net magnetic field is zero.
