2. Find a complex number z such that Re(z^2) + iIm(?(1 + 2i)) = -3. 5. Write the following complex number (cos(?/3) - i sin(?/3))^7 . in the trigonometric form ?(cos + i sin ) and the exponential form ?e^{i?}. 6. Write (?3 + i)^{12} in polar and in Cartesian form. 8. Let z = 2?3 - 2i and w = -32?2 + 32?2i. (a) Sketch the points z and w on the complex plane and convert them to exponential form. (b) Using the exponential form of z and w evaluate ((2?3 - 2i)^{12})/((-32?2 + 32?2i)^4). Simplify your answer to the form a + bi. (c) Determine all values of w^{1/3} and sketch the roots on the complex plane.
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For the equation Re(z) + ilm(z(1 + 2i)) = -3, we can write it as x + i(y(1 + 2i)) = -3, where x and y are the real and imaginary parts of z respectively. Solving this equation gives us x = -3 and y = 0, so the complex number z is -3. Show more…
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