To simplify the expression [tex]\(\left(4^4\right)^{-1}\)[/tex], follow these steps:
1. Apply the Power of a Power property: This property states that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Here, we have [tex]\(a = 4\)[/tex], [tex]\(m = 4\)[/tex], and [tex]\(n = -1\)[/tex].
[tex]\[
\left(4^4\right)^{-1} = 4^{4 \cdot (-1)} = 4^{-4}
\][/tex]
2. Convert to Reciprocal Form: An expression with a negative exponent can be written as the reciprocal of the positive exponent. This means:
[tex]\[
4^{-4} = \frac{1}{4^4}
\][/tex]
Therefore, the simplified form of [tex]\(\left(4^4\right)^{-1}\)[/tex] is [tex]\(\frac{1}{4^4}\)[/tex].
Among the given choices:
- [tex]\(4^3\)[/tex]
- [tex]\(-4^4\)[/tex]
- [tex]\(\frac{1}{4^4}\)[/tex]
- [tex]\(\frac{1}{4^{-4}}\)[/tex]
The correct answer is:
[tex]\(\frac{1}{4^4}\)[/tex]
