- #1
a_skier
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Homework Statement
Prove (a[itex]^{n}[/itex])[itex]^{m}[/itex]=a[itex]^{nm}[/itex]
Homework Equations
Proof by induction
a[itex]^{n}[/itex]*a=a[itex]^{n+1}[/itex]
a[itex]^{n}[/itex]*a[itex]^{m}[/itex]=a[itex]^{nm}[/itex]
The Attempt at a Solution
Let a and n be fixed. I will induct on m.
Suppose m=1. Then a[itex]^{(n)(m)}[/itex]=a[itex]^{n(1)}[/itex]=(a[itex]^{n}[/itex])[itex]^{1}[/itex]
Now assume the hypothesis is true for any integer m in P. I will show this is true for m+1.
a[itex]^{n(m+1)}[/itex]=(a[itex]^{n}[/itex])[itex]^{m+1}[/itex]
Thus the hypothesis is true for m+1.
Is this proof sufficient? I am once again struck by a problem that seems almost too simple to be proved.