Why doesn't $x^0$ equal the same thing as $x\cdot0$?

Question

If $x\cdot0$ means adding zero $x$s together and $x^0$ means multiplying zero $x$s together then conceptually why aren't they both equal to the same thing?

Answer 1

For $x \cdot 0$, it means by definition to "start at $0$, and move $x$ units to the right, $0$ times", which results in $x\cdot 0 = 0$, the starting place.

But for the exponent, we think $x^0$ to mean "start at $1$, and grow (in a compounding fashion) $x$ units in magnitude, $0$ times", which results in $x^ 0 = 1$, the starting place.

As comments pointed out, numbers together with the operation of addition have an additive identity $0$. And numbers together with addition and multiplication have a multiplicative identity $1$. Taking the exponent is shorthand for multiplication, hence why the starting place is 1 (the multiplicative identity).

Answer 2

multiplying exponents with the same base, is the same as adding the exponents, i.e. $a^b * a^c = a^{(b+c)}$

using this consider:

$$ x^0 = x^{-1} x^1 = \frac{1}{x^1}x^1 = 1 $$

OR

$$ x^3 = x^2x \\ x^2 = x^1x \\ x^1 = x^0 x $$

here it becomes clear the only value for $x^0$ could be 1