Created by Andrew Meit, march 6, 2022.

NOTE:UsePowerExpandonFullFormversionstoprovetheProperties
Power[a,p]/Power[a,q]==Power[a,(p-q)]​​a^p/a^q==a^(p-q),Assuming[a,a!=0]​​
p
a
/
q
a
==
p-q
a
(*typeset*)
Power[ab,p]==Power[a,p]Power[b,p]​​(ab)^p==a^pb^p​​
p
(ab)
==
p
a
p
b
(*typeset*)
Power[a/b,p]==Power[a,p]/Power[b,p]​​(a/b)^p==a^p/b^p,Assuming[b,b!=0]​​
p
a
b
==
p
a
/
p
b
(*typeset*)
Power[a,(p/q)]==Power[Power[a,p],1/q]==Power[(Power[a,(1/p)]),p]​​
p/q
a

1/q
(
p
a
)

p
(
1/p
a
)
(*typeset*)​​
p/q
a
==
q
p
a
==
p

q
a

(*typeset*)
Power[Power[a,(1/q)],1/p]==Power[a,(1/(pq))]​​
1/p
(
1/q
a
)
==
1/pq
a
(*typeset*)​​
p
q
a
==
pq
a
(*typeset*)
Power[ab,1/p]==Power[a,1/p]Power[b,1/p]​​
1/p
(ab)
==
1/p
a
1/p
b
(*typeset*)​​
p
ab
==
p
a
p
b
(*typeset*)
Power[a/b,1/p]==Power[a,1/p]/Power[b,1/p],Assuming[b,b!=0]​​
1/p
a
b
==
1/p
a
/
1/p
b
(*typeset*)​​
p
a
b
==
p
a
p
b
(*typeset*)