Question

negative exponents
dividing with like bases
simplify $4^{-4}$.
options: $\frac{1}{16}$, $256$, $\frac{1}{256}$, $16$

Explanation:

Step1: Recall negative exponent rule

The rule for negative exponents is \( a^{-n} = \frac{1}{a^{n}} \) (where \( a
eq0 \) and \( n \) is a positive integer). For \( 4^{-4} \), we first apply the negative exponent rule to \( 4^{-1} \) (maybe there was a typo and it's \( 4^{-4} \) or we simplify step - by - step). Wait, the problem says "Simplify \( 4^{-4} \)"? Wait, the original says "Simplify \( 4^{-4} \)"? Wait, the image says "Simplify \( 4^{-4} \)"? Wait, no, the user's image has "Simplify \( 4^{-4} \)"? Wait, the text in the image: "Simplify \( 4^{-4} \)"? Wait, no, looking at the image again: "Simplify \( 4^{-4} \)"? Wait, the vertical text: "Simplify \( 4^{-4} \)". Let's correct: the rule for negative exponents is \( a^{-n}=\frac{1}{a^{n}} \), so for \( 4^{-4} \), first, \( 4^{-1}=\frac{1}{4} \), but for \( 4^{-4} \), we have \( 4^{-4}=\frac{1}{4^{4}} \).

Step2: Calculate \( 4^{4} \)

\( 4^{4}=4\times4\times4\times4 = 256 \). So \( 4^{-4}=\frac{1}{4^{4}}=\frac{1}{256} \)? Wait, no, wait the options: there is \( \frac{1}{256} \), \( \frac{1}{16} \), \( 256 \), \( 16 \). Wait, maybe the problem is "Simplify \( 4^{- 4} \)"? Wait, no, maybe it's a misprint and it's \( 4^{-(-4)} \)? No, let's re - examine. Wait, the negative exponent rule: \( a^{-n}=\frac{1}{a^{n}} \), so \( 4^{-4}=\frac{1}{4^{4}}=\frac{1}{256} \), but wait, if it's \( 4^{-(-4)} = 4^{4}=256 \). Wait, maybe the problem is "Simplify \( 4^{-4} \)"? Wait, the options: \( \frac{1}{256} \), \( \frac{1}{16} \), \( 256 \), \( 16 \). Let's check \( 4^{-4} \): \( 4^{4}=256 \), so \( 4^{-4}=\frac{1}{256} \). But wait, if the problem was \( 4^{-(-4)}=4^{4} = 256 \), which is one of the options. Wait, maybe the original problem has a typo, and it's \( 4^{-(-4)} \). Let's assume that the problem is to simplify \( 4^{-(-4)} \) (since \( 4^{-4} \) gives \( \frac{1}{256} \), which is an option, and \( 4^{4}=256 \), which is also an option). Wait, the vertical text says "Simplify \( 4^{-4} \)"? No, the user's image: the text is "Simplify \( 4^{-4} \)"? Wait, the OCR: "Simplify 4⁻⁴". So \( 4^{-4}=\frac{1}{4^{4}}=\frac{1}{256} \). But let's check the options. The options are \( \frac{1}{256} \), \( \frac{1}{16} \), \( 256 \), \( 16 \). So if we use the negative exponent rule: \( a^{-n}=\frac{1}{a^{n}} \), so \( 4^{-4}=\frac{1}{4\times4\times4\times4}=\frac{1}{256} \). But wait, maybe the problem was \( 4^{-(-4)} \), which is \( 4^{4}=256 \). Let's check the options again. The option \( 256 \) is there. Wait, maybe the original problem has a sign error. Let's re - derive:

Case 1: Simplify \( 4^{-4} \)
Using \( a^{-n}=\frac{1}{a^{n}} \), \( 4^{-4}=\frac{1}{4^{4}} \)
\( 4^{4}=4\times4\times4\times4 = 256 \), so \( 4^{-4}=\frac{1}{256} \)

Case 2: Simplify \( 4^{-(-4)}=4^{4}=256 \)

Looking at the options, both \( \frac{1}{256} \) and \( 256 \) are present. Wait, maybe the problem is "Simplify \( 4^{-4} \)" and the answer is \( \frac{1}{256} \), or if it's \( 4^{4} \) (due to a double negative), it's \( 256 \). Wait, the vertical text says "Simplify 4⁻⁴", so we go with \( 4^{-4}=\frac{1}{4^{4}}=\frac{1}{256} \). But wait, let's check the options again. The first option on the left is \( \frac{1}{256} \), the middle - top is \( \frac{1}{16} \), middle is \( 256 \), bottom is \( 16 \).

Wait, maybe I made a mistake. Let's re - calculate \( 4^{-4} \):

\( 4^{-1}=\frac{1}{4} \)

\( 4^{-2}=\frac{1}{4^{2}}=\frac{1}{16} \)

\( 4^{-3}=\frac{1}{4^{3}}=\frac{1}{64} \)

\( 4^{-4}=\frac{1}{4^{4}}=\frac{1}{256} \)

So the simplified form of \( 4^{-4} \) is \( \frac{1}{256} \).

Answer:

\(\frac{1}{256}\) (the option with \(\frac{1}{256}\))