Step 1: Identify the complex number expression. Let the given expression be $W$. $$W = \frac{Z-1-i}{3+2i}$$ Step 2: Apply the property of conjugates for division. The conjugate of a quotient is the quotient of the conjugates: $\left(\frac{A}{B}\right)^ = \frac{A^}{B^*}$. So, the conjugate of $W$, denoted as $W^*$, is: $$W^ = \frac{(Z-1-i)^}{(3+2i)^*}$$ Step 3: Find the conjugate of the numerator. The conjugate of a sum or difference is the sum or difference of the conjugates. The conjugate of a real number is itself, and the conjugate of $i$ is $-i$. $$(Z-1-i)^ = Z^ - 1^ - i^ = Z^ - 1 - (-i) = Z^ - 1 + i$$ Step 4: Find the conjugate of the denominator. The conjugate of $a+bi$ is $a-bi$. $$(3+2i)^* = 3 - 2i$$ Step 5: Substitute the conjugates back into the expression for $W^*$. $$W^ = \frac{Z^ - 1 + i}{3 - 2i}$$ Step 6: Rationalize the denominator. Multiply the numerator and the denominator by the conjugate of the denominator, which is $3+2i$. $$W^ = \frac{Z^ - 1 + i}{3 - 2i} \times \frac{3 + 2i}{3 + 2i}$$ Step 7: Multiply the denominators. $$(3 - 2i)(3 + 2i) = 3^2 - (2i)^2 = 9 - 4i^2 = 9 - 4(-1) = 9 + 4 = 13$$ Step 8: Multiply the numerators. $$(Z^ - 1 + i)(3 + 2i) = Z^(3) + Z^*(2i) - 1(3) - 1(2i) + i(3) + i(2i)$$ $$= 3Z^ + 2iZ^ - 3 - 2i + 3i + 2i^2$$ $$= 3Z^ + 2iZ^ - 3 + i + 2(-1)$$ $$= 3Z^ + 2iZ^ - 3 + i - 2$$ $$= 3Z^ + 2iZ^ - 5 + i$$ Step 9: Combine the results to get the final conjugate expression. $$W^ = \frac{3Z^ + 2iZ^* - 5 + i}{13}$$ The conjugate of the expression is: $$\boxed{\frac{3Z^ + 2iZ^ - 5 + i}{13}}$$ Send me the next one 📸