Manipulate[
z=1+I *Pi/n;p=Table[{{0,0},{Re[z^nos], Im[z^nos]},{Re[z^(nos+1)], Im[z^(nos+1)]}},{nos,0,n-1,1}];
Graphics[{
{EdgeForm[Black],Hue[1],Triangle[p]},
Circle[{0,0},1],
Line[{{2.5,0},{-3.5,0}}],
Line[{{2.5,3},{-3.5,3}}],
Text[Style[-1,Large],{-1.15,-.15}],
{Orange,Disk[{-1,0},0.05]},
Text[Style[1,Large],{1.1,-.15}],
Text[Style[0,Large],{0,-.15}],
Text[Style[1+I *Pi/n,Large],{Re[z]+.4, Im[z]+.3}],
{Blue, Disk[{Re[z], Im[z]},.05]},
Text[Style[(1+I *Pi/n)^n,Large],{Re[z^n]-.5, Im[z^n]+.4}],
{Green,Disk[{Re[z^n], Im[z^n]},.05]}
},ImageSize->{600,600}], {n,3,20,1}]

Export["multipli.pdf",%]

(1+2I)*(1.5+I)

Manipulate[Plot[{Exp[x],(1+x/n)^n},{x,-3,2} ,PlotLegends->{Style[StandardForm[e^x],Large],Style[(Collect[StandardForm[1+x/n],x])^n,Large]},ImageSize->{600,600}], {n,3,100,1}]

Nth approximation

Manipulate[N[(1+1/n)^n];N[Expand[(1+x/n)^n]],{n,1,10000,1}]

m=N[Expand[(1+x/7)^7]]
TeXForm[Expand[(1+x/7)^7]]

N[(1+1/3)^3]

Export["exponential1.mov",%],

Powers of complex numbers

Manipulate[
z=a+I *b;p=Table[{{0,0},{Re[z^nos], Im[z^nos]},{Re[z^(nos+1)], Im[z^(nos+1)]}},{nos,0,n-1,1}];
Graphics[{
{EdgeForm[Black],Hue[1],Triangle[p]},
Circle[{0,0},1],
Circle[{-3.5,-3.5},.001],Circle[{3.5,3.5},.001],
Line[{{3.5,0},{-3.5,0}}],
Text[Style[-1,Large],{-1.15,-.15}],
Text[Style[Power ,Large],{2.5,.3}],
Text[Style[n,Large],{3,.3}],
{Orange,Disk[{-1,0},0.05]},
Text[Style[1,Large],{1.1,-.15}],
Text[Style[0,Large],{0,-.15}],
Text[Style[a+b I,Large],{Re[z]+.4, Im[z]+.3}],
{Blue, Disk[{Re[z], Im[z]},.05]},
{Green,Disk[{Re[z^n], Im[z^n]},.05]}
},ImageSize->{800,800}], {n,1,20,1},{{a,0},0,1.30,.02},{{b,1},0.6,1.2,.02}]

Expand[(x^2-1) ((-3+x)^2-4)]//TraditionalForm

Collect[(1+x/4)^4,x]

(Collect[1+x/n,x])^n

Manipulate[
z=a+I *b;p=Table[{{0,0},{Re[z^nos], Im[z^nos]},{Re[z^(nos+1)], Im[z^(nos+1)]}},{nos,0,n-1,1}];
Graphics[{
{EdgeForm[Black],Hue[1],Triangle[p]},
Circle[{0,0},1],
Circle[{-3.5,-3.5},.001],Circle[{3.5,3.5},.001],
Line[{{3.5,0},{-3.5,0}}],
Text[Style[-1,Large],{-1.15,-.15}],
Text[Style[Power ,Large],{2.5,.3}],
Text[Style[n,Large],{3,.3}],
{Orange,Disk[{-1,0},0.05]},
Text[Style[1,Large],{1.1,-.15}],
Text[Style[0,Large],{0,-.15}],
Text[Style[a+b I,Large],{Re[z]+.4, Im[z]+.3}],
{Blue, Rectangle[{Re[z]-.05, Im[z]-.05},{Re[z]+.05, Im[z]+.05}]},
{Green,Rectangle[{Re[z^n]-.05, Im[z^n]-.05},{Re[z^n]+.05, Im[z^n]+.05}]}
},ImageSize->{800,800}], {n,1,20,1},{{a,0},0,1.30,.02},{{b,1},0.6,1.2,.02}]

Manipulate[
z=a+I *b;p=Table[{{0,0},{Re[z^nos], Im[z^nos]},{Re[z^(nos+1)], Im[z^(nos+1)]}},{nos,0,n-1,1}];
Graphics[{
{EdgeForm[Black],Hue[1],Triangle[p]},
Circle[{0,0},1],
Circle[{-3.5,-3.5},.001],Circle[{3.5,3.5},.001],
Line[{{3.5,0},{-3.5,0}}],
Text[Style[-1,Large],{-1.15,-.15}],
{Orange,Disk[{-1,0},0.05]},
Text[Style[1,Large],{1.1,-.15}],
Text[Style[0,Large],{0,-.15}],
Text[Style[n,Bold,Large],{Re[z^n]+.1, Im[z^n]+.1}],
{Blue, Rectangle[{Re[z]-.05, Im[z]-.05},{Re[z]+.05, Im[z]+.05}]},
{Blue,Rectangle[{Re[z^n]-.05, Im[z^n]-.05},{Re[z^n]+.05, Im[z^n]+.05}]}
},ImageSize->{800,800}], {n,1,20,1},{{a,0.932},-1.3,1.30,.02},{{b,0.36`
},-1.2,1.2,.02}]

Manipulate[N[(1+I Pi/m)^m],{m,1
,100,1}]