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In Problems 16, put the complex number in the form a + ib. In Problems 712, find the roots of the given equation. In Problems 1318, put the complex number into the polar form r cis θ. In Problems 1924, use the polar form to simplify the given expression. In Problems 2528, compute both square roots of the given complex number using the polar form. In Problems 2932, put the given exponent in the form a + ib. In Problems 3336, find the derivative. In Problems 3740, find the general solution of the complex differential equation. In Problems 4144, solve the given complex initial value problem. 46 Prove that the conjugate of a complex number r cis θ is r cis (θ). 47 Prove that for any nonzero complex number z, , where θ is the argument of z. 48 Prove that for any two complex numbers u and z, the sum of the conjugates of u and z is equal to the conjugate of the sum of u and z, and similarly for products. In symbols, 49 Prove that for any two complex numbers u and z, 50 Use De Moivre's Formula with n = 2, cos (2θ) + i sin (2θ) = (cos θ + i sin θ)^{2}, to obtain expressions for cos (2θ) and sin (2θ) in terms of cos θ and sin θ. 51 Use De Moivre's Formula with n = 3, cos (3θ) + i sin (3θ) = (cos θ + i sin θ)^{3}, to obtain expressions for cos (3θ) and sin (3θ) in terms of cos θ and sin θ. 52 Find the solution of the initial value problem z' + (a + ib)z = 0, z(0) = e^{c+id }where a, b, c, and d are real numbers. 53 Show that every solution of the differential equation z' + ibz = 0 has constant absolute value (where b is a real number).


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