To find the equivalent expression of [tex]\( 4^{-9} \)[/tex], we need to simplify it using the properties of exponents. Specifically, we will use the property that a negative exponent indicates a reciprocal.
Given the expression [tex]\( 4^{-9} \)[/tex], a negative exponent means that the base (in this case, 4) should be taken as the reciprocal raised to the positive exponent. This gives us:
[tex]\[ 4^{-9} = \frac{1}{4^9} \][/tex]
In this way, [tex]\( 4^{-9} \)[/tex] simplifies to [tex]\( \frac{1}{4^9} \)[/tex].
Now, let's compare this result with the given options:
1. [tex]\(-\frac{1}{4^9}\)[/tex]
2. [tex]\(\frac{1}{9^4}\)[/tex]
3. [tex]\(\frac{1}{4^9}\)[/tex]
4. [tex]\(-\frac{1}{9^4}\)[/tex]
From our calculation, we can see that the correct equivalent expression matches the third option:
[tex]\[ \frac{1}{4^9} \][/tex]
Thus, the equivalent expression of [tex]\( 4^{-9} \)[/tex] is:
[tex]\[ \frac{1}{4^9} \][/tex]
