negative exponents\ndividing with like bases\nsimplify $4^{-4}$.\noptions: $\frac{1}{16}$, $256$…

Explanation:

Step1: Recall negative exponent rule

The rule for negative exponents is ( a^{-n} = \frac{1}{a^{n}} ) (where ( a\neq0 ) 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}))