The question is ambiguous: xa+1 * xa+1 = x2(a+1)

or

(xa + 1)(xa + 1) = x2a + 2xa + 1

0.75

You first define negative powers as the reciprocals of the positive powers ie x-a = 1/xa.

You have the folowing property for positive powers: xa * xb = xa+b

You extend the following property to negative powers:

So xa * x-a = x0. But by definition, xa * x-a = xa * 1/xa = 1

So x0 = 1

Any number raised to the power 0 is 1.

This follow from the law of multiplications of power:

xa * xb = xa+b

Now, if you put b = 0, you get xa + x0 = xa+0

and since a+0 = a, the right hand side is xa.

So you have xa * x0 = xa

and using the property of the multiplicative identity, xa = 1.

The multiplicative law of indices states that

xa * xb = xa+b

Now, if you put b = 0 in that equation you get

xa * x0 = xa+0

But a+0 = a so the right hand side is simply xa

Which means, the equation becomes

xa * x0 = xa

This is true for any x.

That is, multiplying any number by x0 leaves it unchanged.

By the identity property of multiplication, there is only one such number and that is 1.

So x0 must be 1.