Use la forma polar de los números complejos dados para hacer las operaciones que se piden: a) Para \( z=1-0.5 \mathrm{i} \) y \( w=3-7 \mathrm{i} \) calcule \( z w \). b) Para \( z=-36 \mathrm{i} \) y \( w=-2-i \) calcule \( z w \).

Alright, I need to solve these complex number multiplication problems using their polar forms. Let's tackle them one by one. **Problem a:** Given \( z = 1 - 0.5i \) and \( w = 3 - 7i \), calculate \( z \times w \). First, I'll convert both complex numbers to their polar forms. The polar form of a complex number \( z = a + bi \) is given by \( z = r(\cos \theta + i \sin \theta) \), where \( r = \sqrt{a^2 + b^2} \) and \( \theta = \arctan\left(\frac{b}{a}\right) \). **For \( z = 1 - 0.5i \):** - \( r_z = \sqrt{1^2 + (-0.5)^2} = \sqrt{1 + 0.25} = \sqrt{1.25} \approx 1.118 \) - \( \theta_z = \arctan\left(\frac{-0.5}{1}\right) = \arctan(-0.5) \approx -26.565^\circ \) or \( 333.435^\circ \) (since it's in the fourth quadrant) **For \( w = 3 - 7i \):** - \( r_w = \sqrt{3^2 + (-7)^2} = \sqrt{9 + 49} = \sqrt{58} \approx 7.62 \) - \( \theta_w = \arctan\left(\frac{-7}{3}\right) \approx -66.8^\circ \) or \( 293.2^\circ \) (since it's in the fourth quadrant) Now, to multiply two complex numbers in polar form, I multiply their magnitudes and add their angles: - \( r_{zw} = r_z \times r_w = 1.118 \times 7.62 \approx 8.5 \) - \( \theta_{zw} = \theta_z + \theta_w = -26.565^\circ + (-66.8^\circ) = -93.365^\circ \) So, \( z \times w \) in polar form is approximately \( 8.5(\cos(-93.365^\circ) + i \sin(-93.365^\circ)) \). To convert this back to rectangular form: - \( \cos(-93.365^\circ) \approx -0.052 \) - \( \sin(-93.365^\circ) \approx -0.998 \) - Therefore, \( z \times w \approx 8.5 \times (-0.052) + 8.5 \times (-0.998)i \approx -0.442 - 8.483i \) **Problem b:** Given \( z = -36i \) and \( w = -2 - i \), calculate \( z \times w \). First, let's express both complex numbers in standard form: - \( z = 0 - 36i \) - \( w = -2 - i \) Now, multiply them: - \( z \times w = (0 - 36i)(-2 - i) = 0 \times (-2) + 0 \times (-i) + (-36i) \times (-2) + (-36i) \times (-i) \) - \( = 0 + 0 + 72i + 36i^2 \) - Since \( i^2 = -1 \), \( 36i^2 = 36 \times (-1) = -36 \) - So, \( z \times w = 72i - 36 \) Therefore, \( z \times w = -36 + 72i \).