To solve the given expression [tex]\( 4^2 \div 4^6 \)[/tex], we can use the properties of exponents, specifically, the rule for dividing powers with the same base. The rule is:
[tex]\[ \frac{a^m}{a^n} = a^{m-n} \][/tex]
Here, we have the base [tex]\(4\)[/tex] and the exponents [tex]\(2\)[/tex] and [tex]\(6\)[/tex]. According to the rule, we subtract the exponents:
[tex]\[ 4^2 \div 4^6 = 4^{2-6} \][/tex]
Simplifying the exponents:
[tex]\[ 4^{2 - 6} = 4^{-4} \][/tex]
So, the simplified expression is [tex]\( 4^{-4} \)[/tex].
Next, we can determine the value of [tex]\( 4^{-4} \)[/tex]. Recall that for a negative exponent, we use the rule:
[tex]\[ a^{-n} = \frac{1}{a^n} \][/tex]
Applying this rule:
[tex]\[ 4^{-4} = \frac{1}{4^4} \][/tex]
Now, let's calculate [tex]\( 4^4 \)[/tex]:
[tex]\[ 4^4 = 4 \times 4 \times 4 \times 4 = 256 \][/tex]
Therefore:
[tex]\[ 4^{-4} = \frac{1}{256} \][/tex]
Given the numerical value, [tex]\( \frac{1}{256} \)[/tex] is equivalent to:
[tex]\[ 0.00390625 \][/tex]
So, the correct answer and the simplified form of [tex]\( 4^2 \div 4^6 \)[/tex] is [tex]\( 4^{-4} \)[/tex], which equals [tex]\( 0.00390625 \)[/tex].
Thus, the correct selection from the given options is [tex]\( \boxed{4^{-4}} \)[/tex].
