$\frac{12 \operatorname{cis}\left(\frac{8}{3} \pi\right)}{11 \operatorname{cis}\left(\frac{12}{11} \pi\right)}$

Step 1 ：The given expression is \(\frac{12 \operatorname{cis}\left(\frac{8}{3} \pi\right)}{11 \operatorname{cis}\left(\frac{12}{11} \pi\right)}\).

Step 2 ：The polar form of a complex number is \(r(\cos(\theta) + i\sin(\theta))\), which can also be written as \(r \operatorname{cis}(\theta)\).

Step 3 ：The division of two complex numbers in polar form is done by dividing the magnitudes and subtracting the angles.

Step 4 ：So, we need to divide 12 by 11 to get the magnitude of the result, and subtract \(\frac{12}{11} \pi\) from \(\frac{8}{3} \pi\) to get the angle of the result.

Step 5 ：Calculating the magnitude, we get approximately 1.09.

Step 6 ：Calculating the angle, we get \(\frac{52}{33} \pi\).

Step 7 ：Thus, the simplified form of the given expression is \(1.09 \operatorname{cis}\left(\frac{52}{33} \pi\right)\).

Step 8 ：Final Answer: \(\boxed{\frac{12 \operatorname{cis}\left(\frac{8}{3} \pi\right)}{11 \operatorname{cis}\left(\frac{12}{11} \pi\right)} = 1.09 \operatorname{cis}\left(\frac{52}{33} \pi\right)}\)