First convert the complex numbers into a+ib form:
z₁=1+i/(√3-i)=1+i(√3+i)/(3-i²)=1+¼i√3-¼=¾+¼i√3.
z̄₁=¾-¼i√3; |z₁|=√((9/16)+(3/16))=½√3.
z₂=(e^i)+π/2 (assumed) = π/2+cos(1)+isin(1). The angle is radians.
cos(1)=0.5403, sin(1)=0.8415 approx. π/2=1.5708 approx.
z₂=2.1111+0.8415i approx; z̄₂=2.1111-0.8415i.
z₃=2√3-2i; z̄₃=2√3+2i.
(a) z̄₁+z₃=¾-¼i√3+2√3-2i=(¾+2√3)-(2+¼√3)i=4.2141-2.4330i.
(b) z₁-z̄₂=¾+¼i√3-(2.1111-0.8415i)=-1.3611-0.4085i.
|z₁|+z₃=½√3+2√3-2i=5(√3)/2-2i;
1/(|z₁|+z₃)=(5(√3)/2+2i)/(91/4)=(10√3+8i)/91.
(z₁-z̄₂)/(|z₁|+z₃)=-(1.3611+0.4085i)(10√3+8i)/91=
-(1/91)(23.5749+17.9635i-3.2677)=-(1/91)(20.3073+17.9635i)=
-0.2232-0.1974i.
(c) z₁⁵+z₃⁴.
z₃ can be expressed 4e^(πi/6), so z₃⁴=256e^(⅔πi)=
256(cos(⅔π)+isin(⅔π))=256(-½+i√3/2)=
-128+128i√3.
z₁ can be expressed (√3/2)(cos30°+isin30°)=(√3/2)e^(πi/6).
z₁⁵=9√3/32e^(⅚πi)=(9√3/32)(-cos30°+isin30°)=
-27/64+9i√3/64.
z₁⁵+z₃⁴=-27/64+9i√3/64-128+128i√3=
-8219/64+221.9461i approx.
(d) z₁²=9/16-3/16+3i√3/8=⅜+⅜i√3.
1/z₂=1/(π/2+cos(1)+isin(1))=
(π/2+cos(1)-isin(1))/((π/2+cos(1))²+sin²(1))=
(π/2+cos(1)-isin(1))/(π²/4+πcos(1)+1)=
(π/2+cos(1)-isin(1))/5.1648=
0.4087-0.1629i.
1/z₂³=0.4087³-3(0.4087)²(0.1629)i+3(0.4087)(0.1629)²i²-(0.1629)³i³
0.0683-0.0817i-0.0325+0.0043i=
0.0357-0.0773i approx.
z₁²/z₂³=(⅜+⅜i√3)(0.0357-0.0773i)=
0.0134-0.0290i+0.0232-0.0502=0.0366-0.0792i.
(e) z₂+z₁²=2.1111+0.8415i+⅜+⅜i√3=
2.4861+1.4910i.
|z₂+z₁²|=√(2.4861²+1.4910²)=2.8989.