z=2-4i compute z^10

Question

complex number problem z=2-4i compute z^10

Answer 1

METHOD 1 (long)

z=2-4i=2(1-2i); z¹⁰=2¹⁰(1-2i)¹⁰.

Coefficients in expansion:

1, 10, 45, 120, 210, 252,  210, 120, 45, 10, 1.

(1-2i)¹⁰=

1-10(2i)+45(2i)²-120(2i)³+210(2i)⁴-252(2i)⁵+210(2i)⁶-120(2i)⁷+45(2i)⁸-10(2i)⁹+(2i)¹⁰,

(1-2i)¹⁰=

1+45(2i)²+210(2i)⁴+210(2i)⁶+45(2i)⁸+(2i)¹⁰ -

(10(2i)+120(2i)³+252(2i)⁵+120(2i)⁷+10(2i)⁹),

(1-2i)¹⁰=1-45(2²)+210(2⁴)-210(2⁶)+45(2⁸)-2¹⁰ -

i(10(2)-120(2³)+252(2⁵)-120(2⁷)+10(2⁹)),

(1-2i)¹⁰=1+45(2⁸-2²)-210(2⁶-2⁴)-2¹⁰ -

i(10(2+2⁹)-120(2³+2⁷)+252(2⁵)),

(1-2i)¹⁰=1+45(252)-210(48)-1024 -

i(10(514)-120(136)+252(32)),

(1-2i)¹⁰=1+11340-10080-1024-i(5140-16320+8064),

(1-2i)¹⁰=237+3116i.

z¹⁰=1024(237+3116i)=242688+3190784i.

METHOD 2 (short)

z=2-4i=re^iθ=r(cosθ+isinθ); z¹⁰=r¹⁰e^10iθ=r¹⁰(cos(10θ)+isin(10θ)).

rcosθ=2, rsinθ=-4; r²=20, tanθ=-2, θ=-1.10715 radians approx, 10θ=-11.0715 radians.

z¹⁰=20⁵(0.07584+0.99712i)=242688+3190784i.

The two methods concur in their results (using a calculator for accuracy in Method 2).