Certainly! Let's simplify the expression [tex]\( p^0 \)[/tex].
In mathematics, any non-zero number raised to the power of 0 is defined to be 1. This is a fundamental principle of exponents.
There are a few ways to understand why this is the case:
1. Definition: By definition, for any non-zero number [tex]\( p \)[/tex], the expression [tex]\( p^n \)[/tex] represents [tex]\( p \)[/tex] multiplied by itself [tex]\( n \)[/tex] times. When [tex]\( n = 0 \)[/tex], we have an empty product of [tex]\( p \)[/tex], and the convention is to define any number to the power of 0 as 1.
2. Exponential Laws: Considering the laws of exponents, specifically the property that [tex]\( p^a \cdot p^b = p^{a+b} \)[/tex], let's set [tex]\( a \)[/tex] and [tex]\( b \)[/tex] to 0. We get:
[tex]\[
p^0 \cdot p^0 = p^{0+0} = p^0.
\][/tex]
To maintain consistency, we see that [tex]\( p^0 \)[/tex] must equal 1 because multiplying any number by 1 does not change its value.
3. Sequential Reasoning: Take a series of decreasing exponents, for instance, [tex]\( p^3, p^2, p^1 \)[/tex]. Dividing each by [tex]\( p \)[/tex], we have:
[tex]\[
p^3 = p \cdot p \cdot p,
\][/tex]
[tex]\[
p^2 = p \cdot p,
\][/tex]
[tex]\[
p^1 = p.
\][/tex]
Continuing the pattern, dividing [tex]\( p^1 \)[/tex] by [tex]\( p \)[/tex] once more gives:
[tex]\[
p^0 = \frac{p^1}{p} = \frac{p}{p} = 1.
\][/tex]
Therefore, the simplified form of [tex]\( p^0 \)[/tex] is indeed:
[tex]\[
1
\][/tex]
So, the correct and simplified form of [tex]\( p^0 \)[/tex] is:
[tex]\[
\boxed{1}
\][/tex]
