1) (-2 -3i)⁵ = -(2+3i)⁵ = -( 2⁵ +5*2⁴*3i +10*2³*(3i)²+10*2²*(3i)³ +5*2*(3i)⁴ +(3i)⁵ ) =
-(32 +240i -720 -1080 i +810 +243 i) = -122 +597 i .
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2) (-1+i)/√2*e^(π/3*i) =√2(cos3π/4+isin3π/4) /√2*e^(π/3*i) =e^(3π/4*i)/e^(π/3*i) =
e^ ((3π/4 -π/3) )*I =e ^(5π/12*i ).
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(-1+i)/√2*e^(π/3*i) =(-1+i)/(√2(cosπ/3+ icosπ/3) =(-1+i)/√2(1/2 +√3/2i) =
√2(-1+i) /(1+i√3) =√2(-1+i) (1-i√3) /(1+i√3)(1-i√3)=√2(-1+i) (1-i√3) /4 =...
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3) (3i+3)/2i¹⁰ =(3+3i) / 2*(i²)⁵ =(3+3i) / 2*(-1)⁵ = - 3/2(1+i) =
- 3/2*√2(cosπ/4+sinπ/4i) = -(3√2)/2(cosπ/4+sinπ/4i).
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4) 4(cosπ/6 + i sinπ/6) = 4(√3 /2 + i* 1/2) =2√3 +2i .