i . ) z 1 = 3 + i , β β z 2 = 2 β i z 1 z 2 = ( 3 + i ) ( 2 β i ) = 6 β 3 i + 2 i + 1 = 7 β i β£ z 1 z 2 β£ = 7 2 + ( β 1 ) 2 = 50 = 5 2 arg β‘ ( β£ z 1 z 2 β£ ) = arctan β‘ ( β 1 7 ) = 3 Ο 2 + Ο 2 β 0.142 = 6.141 i i . ) z 1 z 2 = 50 e 6.141 i z 1 z 2 = 50 ( cos β‘ ( 6.141 ) + i sin β‘ ( 6.141 ) ) i i i . ( z 1 z 2 ) 4 = 2500 ( cos β‘ ( 6.141 Γ 4 ) + i sin β‘ ( 6.141 Γ 4 ) ) = 2500 ( cos β‘ ( 25.56 ) + i sin β‘ ( 25.56 ) ) \displaystyle
i.)\\
z_1 = 3 + i,\,\, z_2 = 2 - i\\
z_1 z_2 = (3 + i)(2 - i) = 6 - 3i + 2i + 1 = 7 - i\\
|z_1 z_2| = \sqrt{7^2 + (-1)^2} = \sqrt{50} = 5\sqrt{2}\\
\arg(|z_1 z_2|) = \arctan\left(\frac{-1}{7}\right) = \frac{3\pi}{2} + \frac{\pi}{2} - 0.142
= 6.141\\
ii.)\\
z_1 z_2 = \sqrt{50}e^{6.141i}\\
z_1 z_2 = \sqrt{50}(\cos(6.141) + i\sin(6.141))\\
iii. \\
(z_1 z_2)^4 = 2500(\cos(6.141\times 4) + i\sin(6.141\times 4)) = 2500(\cos(25.56) + i\sin(25.56)) i . ) z 1 β = 3 + i , z 2 β = 2 β i z 1 β z 2 β = ( 3 + i ) ( 2 β i ) = 6 β 3 i + 2 i + 1 = 7 β i β£ z 1 β z 2 β β£ = 7 2 + ( β 1 ) 2 β = 50 β = 5 2 β arg ( β£ z 1 β z 2 β β£ ) = arctan ( 7 β 1 β ) = 2 3 Ο β + 2 Ο β β 0.142 = 6.141 ii . ) z 1 β z 2 β = 50 β e 6.141 i z 1 β z 2 β = 50 β ( cos ( 6.141 ) + i sin ( 6.141 )) iii . ( z 1 β z 2 β ) 4 = 2500 ( cos ( 6.141 Γ 4 ) + i sin ( 6.141 Γ 4 )) = 2500 ( cos ( 25.56 ) + i sin ( 25.56 ))
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