Answer:
In rectangular form, a complex number is expressed as $x + yi$, where $x$ and $y$ are the real and imaginary parts, respectively. The conversion from polar form to rectangular form is done using the following equations:
$x = r \cos \theta$
$y = r \sin \theta$
where $r$ is the magnitude (or modulus) of the complex number and $\theta$ is the argument (or phase) of the complex number.
In the given problem, we are given the complex number $z = 3(cos(7pi/6)+ i sin (7pi/6))$. We can write this in polar form as $z = 3 \cdot cis(7pi/6)$, where $cis(x) = cos(x) + i \cdot sin(x)$. Thus, we have $r = 3$ and $\theta = 7pi/6$. Using the equations above, we can convert this to rectangular form to obtain:
$x = r \cos \theta = 3 \cos (7pi/6) = -\frac{3}{2}$
$y = r \sin \theta = 3 \sin (7pi/6) = \frac{\sqrt{3}}{2}$
Therefore, the rectangular form of the complex number is $-\frac{3}{2} + \frac{\sqrt{3}}{2} i$.