Why is X raised to the power of 0 = 1?

Question

So as the topic states, why is 5^0 = 1 and not 5 or 0? Is it only because the other exponential laws wouldn't work if it was not the case?

Answer 1

This convention allows us to use the additive property of exponents: $b^xb^y = b^{x+y}$ without having to define an edge case for $0$. Example:

$$2^4 = 16 \\ 2^{-4} = \frac{1}{16}\\ 1=16\cdot \frac{1}{16} = 2^4\cdot 2^{-4} = 2^{4-4} = 2^0.$$

Answer 2

To Prove:- $x^0=1$

We know that $x^0 = x \times x$ ($0$ Times)

we all know $1-1=0$

so we can say that $x^0 = x^{1-1}$

Now We Got $x^0=(x^1) \times (x^{-1})$

$x^0=x^1\times {1\over x^1}$ where $x^1$ and $x^1$ gets cancelled

Now we got $x^0=1$ Hence Proved

Answer 3

$5^0$ is the product of zero $5$s, which is the empty product, and there are plenty of reasons to define that as $1$.