To simplify any of these expressions, it is critical that we remember an exponent only applies to what it is touching or immediately next to.

1. In the expression $\left(173 x^4 y^{251}\right)^0\text{,}$ the exponent $0$ applies to everything inside the parentheses.
   $$\left(173 x^4 y^{251}\right)^0 = 1$$
2. In the expression $(-8)^0$ the exponent applies to everything inside the parentheses, $-8\text{.}$
   $$(-8)^0 = 1$$
3. In contrast to the previous example, the exponent only applies to the $8\text{.}$ The exponent has a higher priority than negation in the order of operations. We should consider that $-8^0 = -\mathopen{}\left(8^0\right)\mathclose{}\text{,}$ and so:
   $$\begin{aligned} -8^0 \amp= -\mathopen{}\left(8^0\right)\mathclose{}\\ \amp= -1 \end{aligned}$$
4. In the expression $3x^0\text{,}$ the exponent $0$ only applies to the $x\text{:}$
   $$\begin{aligned} 3x^0 \amp= 3\cdot x^0\\ \amp= 3\cdot 1\\ \amp= 3 \end{aligned}$$