I would apply definition.

https://en.wikipedia.org/wiki/Kronecker_delta
I don't know much about the domain of Kronecker Deltas, but gave it an old college try.

For example, for evaluating (1) I'd use the piecewise definition, and extend when taking it to the power j.

There are four cases now for our new piecewise definition.

When both i and j not equal to 0. Result: 0

When i not equal to 0, j equal to 0. Result: Undefined / 0^0

When i equal to 0, j not equal to 0. Result: 1

When both i and j equal to 0. Result: 0

Do the same for delta j to the power of i.

When both i and j not equal to 0. Result: 0

When i not equal to 0, j equal to 0. Result: 1

When i equal to 0, j not equal to 0. Result: Undefined

When both i and j equal to 0. Result: 0

Take the product of these two expressions and you'll get

When both i and j not equal to 0. Result: 0

When i not equal to 0, j equal to 0. Result: Undefined

When i equal to 0, j not equal to 0. Result: Undefined

When both i and j equal to 0. Result: 0