Exponent Laws

Author

Eric W. Weisstein
July 29, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/ExponentLaws.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ExponentLaws.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Real

FullSimplify[(x^m)(x^n)]

FullSimplify[((x^m)/(x^n))]

FullSimplify[((((x^m)))^n)]

FullSimplify[((((x^m)))^n)-(x^(m n)),(m|n)∈Reals&&x>0]

FullSimplify[(((x y))^m)]

FullSimplify[(((x y))^m)-(x^m) (y^m)]

FullSimplify[(((x y))^m)-(x^m) (y^m),(m|n)∈Reals&&x>0]

FullSimplify[(((x y))^m)-(x^m) (y^m),(m|n)∈Reals&&y>0]

FullSimplify[(((x y))^m)-(x^m) (y^m),(m|n)∈Reals&&(y>0||x>0)]

Table[(((x y))^m)-(x^m) (y^m)/.{x->Random[Real,{0,10},20],y->Random[Complex,(I+1)2{-1,1},20],m->Random[Real,{-10,10},20],n->Random[Real,{-10,10},20]},{50}]//Chop//Union

Table[(((x y))^m)-(x^m) (y^m)/.{y->Random[Real,{0,10},20],x->Random[Complex,(I+1)2{-1,1},20],m->Random[Real,{-10,10},20],n->Random[Real,{-10,10},20]},{50}]//Chop//Union

FullSimplify[((((x/y)))^n)]

FullSimplify[((((x/y)))^n)-((x^n)/(y^n))]

FullSimplify[(x^(-n))]

FullSimplify[(x^(-n))-(1/(x^n))]

FullSimplify[((((x/y)))^(-n))]

FullSimplify[((((x/y)))^(-n))-((((y/x)))^n)]

Complex

PowerExpand[z^(a I),Assumptions->True]

TrigToExp[ComplexExpand[z^(a I)]]

Select[Table[{{z,a},z^(a I),(E^(-a Arg[z])) ((((z^2)))^((I a)/2)),((Abs[z])^(I a))}/.{z->RandomReal[{-1,1}],a->RandomReal[{-1,1}]},{10}],Chop[Subtract@@Take[#,-2]]≠0&]

{z^(a I),(E^(-a Arg[z])) ((((z^2)))^((I a)/2)),((Abs[z])^(I a))}/.Thread[{z,a}->{-0.26584563340432155`,0.12760430981628224`}]

Length[%]

PowerExpand[(x*y)^a]

PowerExpand[(x*y)^a,Assumptions->True]

PowerExpand[(x/y)^a,Assumptions->True]

Along the Real Axis

Plot3D[Re[(Sqrt[x+I y])(Sqrt[(1/(x+I y))])],{x,-10,10},{y,-10,10},PlotPoints->50,Mesh->False]

Simplify[(Sqrt[x])(Sqrt[(1/x)]),#]&/@{x<0,Re[x]>0}

<<Graphics`Colors`

Plot[(Sqrt[x])(Sqrt[(1/x)]),{x,-10,10},PlotStyle->Red]

{(((-1))^(1/2)),((((1/(-1))))^(1/2))}//N

{(((-1))^(1/2)),Sqrt[-1],((((1/(-1))))^(1/2)),(Sqrt[(1/(-1))])}

{(((-1))^(1/2.)),Sqrt[-1],((((1/(-1))))^(1/2.)),(Sqrt[(1/(-1))])}//Chop

FullSimplify[(Sqrt[x])(Sqrt[(1/x)])]

Reduce[(Sqrt[x])(Sqrt[(1/x)])!=1,x,Complexes]