Question

If an expression was (x * y)^n , so (x times y) to the power of n, would you multiply x and y first or distribute the exponent, making (x^n * y^n)?

Answer 1

Either order is correct.

Small example: let's take
`n=3`. Then


`(x* y)^3=(x* y)*(x* y)*(x* y)`

By the commutative property, we can write


`(x* y)^3=(x* y)*\underbrace{(y* x)}*(x* y)`

By the associative property, we can regroup consecutive terms.


`(x* y)^3=(x*(y* y))*((x* x)* y)`

`(x* y)^3=(x* y^2)*(x^2* y)`

By the associative property again, we can regroup terms and write


`(x* y)^3=x*(y^2* x^2)* y`

Commutativity:


`(x* y)^3=x*(x^2* y^2)* y`

Associativity:


`(x* y)^3=(x* x^2)*(y^2* y)`

`(x* y)^3=x^3* y^3`

You can show by induction that this holds in general for
`n` so that
`(x* y)^n=x^n* y^n`.