Use De Moivre's theorem to find the indicated power. Write the result in rectangular form, $a+b \bar{i}$. Give an exact answer. \[ \left[4\left(\cos \frac{\pi}{36}+i \sin \frac{\pi}{36}\right)\right]^{6}= \]

Step 1 ：Given the expression \(\left[4\left(\cos \frac{\pi}{36}+i \sin \frac{\pi}{36}\right)\right]^{6}\)

Step 2 ：We can use De Moivre's theorem to simplify the expression. De Moivre's theorem states that for any real number x and integer n, \((\cos(x) + i*\sin(x))^n = \cos(nx) + i*\sin(nx)\)

Step 3 ：In this case, we have x = \(\frac{\pi}{36}\) and n = 6. So we can apply De Moivre's theorem to get \(\cos(6*\frac{\pi}{36}) + i*\sin(6*\frac{\pi}{36})\)

Step 4 ：Calculating the above expression, we get a complex number z = (3.984778792366982+0.34862297099063266j)

Step 5 ：Raising z to the power of 6, we get z_power_6 = (3547.240053901062+2048j)

Step 6 ：Thus, the result in rectangular form, a+b \(\bar{i}\), is \(\boxed{3547.24 + 2048 \bar{i}}\)