1. The problem is to simplify the expression $Z=\frac{4i}{1 - i\sqrt{3}}$. 2. To simplify a complex fraction, multiply numerator and denominator by the conjugate of the denominator. 3. The conjugate of $1 - i\sqrt{3}$ is $1 + i\sqrt{3}$. 4. Multiply numerator and denominator: $$Z = \frac{4i}{1 - i\sqrt{3}} \times \frac{1 + i\sqrt{3}}{1 + i\sqrt{3}} = \frac{4i(1 + i\sqrt{3})}{(1 - i\sqrt{3})(1 + i\sqrt{3})}$$ 5. Expand numerator: $$4i \times 1 = 4i$$ $$4i \times i\sqrt{3} = 4i^2 \sqrt{3} = 4(-1)\sqrt{3} = -4\sqrt{3}$$ So numerator is $4i - 4\sqrt{3}$. 6. Expand denominator using difference of squares: $$(1)^2 - (i\sqrt{3})^2 = 1 - (i^2)(3) = 1 - (-1)(3) = 1 + 3 = 4$$ 7. So the expression becomes: $$Z = \frac{4i - 4\sqrt{3}}{4} = \frac{4i}{4} - \frac{4\sqrt{3}}{4} = i - \sqrt{3}$$ 8. Final simplified form is: $$Z = i - \sqrt{3}$$