Let $a,b,n$ be any integers, show that: $x^{n}\cdot{y^{n}}=(xy)^{n}$

Question

I know the rule of adding exponents while multiplying same integers: $$x^{a}\cdot{x^{b}}=x^{a+b}$$ but that doesn't seem to help, in this case. An elementary proof would be appreciated.

Regards

Answer 1

Hint: Show by induction on $n$ that if $x,y$ are any integers you have $$x^ny^n=(xy)^n$$

The inductive step boils down to $y^nx=xy^n$.

Answer 2

$$(xy)^n= \underbrace { (xy)\cdots (xy)}_{n\text{(xy)'s}} = \underbrace {x\cdots x}_{n\text{x's}} \times \underbrace {y\cdots y}_{n\text{y's}}=x^ny^n$$

This is easy enough, so induction is not necessary.

Answer 3

expand the RHS

     (xy)^n = (xy)(xy)(xy)...(xy)

                  n times

Now since x,y integers, they commute, change order

         = x.x.x.x...y.y.y.y..

            n times   n times

Hence

         = x^n.y^n