To determine which expressions are equivalent to [tex]\( 4^{-2} \cdot 7^{-2} \)[/tex], we can use the rules of exponents to simplify and compare the given expressions.
1. Start with the original expression:
[tex]\[ 4^{-2} \cdot 7^{-2} \][/tex]
This can be rewritten using the property of exponents that states [tex]\( a^{-m} = \frac{1}{a^m} \)[/tex]:
[tex]\[ 4^{-2} \cdot 7^{-2} = \frac{1}{4^2} \cdot \frac{1}{7^2} = \frac{1}{4^2 \cdot 7^2} \][/tex]
Since [tex]\( 4^2 \cdot 7^2 = (4 \cdot 7)^2 = 28^2 \)[/tex], we get:
[tex]\[ \frac{1}{4^2 \cdot 7^2} = \frac{1}{28^2} \][/tex]
Now, let's evaluate each given expression to see if it matches [tex]\( \frac{1}{28^2} \)[/tex].
### Expression A: [tex]\((4 \cdot 7)^{-4}\)[/tex]
We can rewrite [tex]\((4 \cdot 7)^{-4}\)[/tex] using the negative exponent rule:
[tex]\[ (4 \cdot 7)^{-4} = \frac{1}{(4 \cdot 7)^4} \][/tex]
However, [tex]\((4 \cdot 7)^4 = 28^4\)[/tex], so:
[tex]\[ \frac{1}{28^4} \][/tex]
Thus, [tex]\((4 \cdot 7)^{-4} \neq \frac{1}{28^2}\)[/tex].
### Expression B: [tex]\(\frac{1}{28^2}\)[/tex]
This is already in the simplified form that matches [tex]\(\frac{1}{28^2}\)[/tex], so:
[tex]\[ \frac{1}{28^2} = \frac{1}{28^2} \][/tex]
Thus, this expression is equivalent.
### Expression C: [tex]\(\frac{7^{-2}}{4^2}\)[/tex]
We rewrite the numerator using the negative exponent rule:
[tex]\[ \frac{7^{-2}}{4^2} = \frac{1}{7^2 \cdot 4^2} \][/tex]
However, this becomes:
[tex]\[ \frac{1}{7^2 \cdot 4^2} = \frac{1}{(7 \cdot 4)^2} = \frac{1}{28^2} \][/tex]
Thus, this matches our original expression and is equivalent.
### Expression D: [tex]\((4 \cdot 7)^4\)[/tex]
This can be directly calculated as:
[tex]\[ (4 \cdot 7)^4 = 28^4 \][/tex]
Clearly, [tex]\((4 \cdot 7)^4 \neq \frac{1}{28^2}\)[/tex].
Based on the evaluations, the expressions equivalent to [tex]\( 4^{-2} \cdot 7^{-2} \)[/tex] are:
B) [tex]\( \frac{1}{28^2} \)[/tex] and C) [tex]\( \frac{7^{-2}}{4^2} \)[/tex]
