Let's determine which expressions are equivalent to [tex]\(4^{-2} \cdot 7^{-2}\)[/tex] through a step-by-step solution.
First, let's evaluate [tex]\(4^{-2}\)[/tex]:
[tex]\[
4^{-2} = \left(\frac{1}{4}\right)^2 = \frac{1}{4^2} = \frac{1}{16} = 0.0625
\][/tex]
Next, let's evaluate [tex]\(7^{-2}\)[/tex]:
[tex]\[
7^{-2} = \left(\frac{1}{7}\right)^2 = \frac{1}{7^2} = \frac{1}{49} \approx 0.02040816326530612
\][/tex]
Now, multiply these two values together:
[tex]\[
4^{-2} \cdot 7^{-2} = 0.0625 \cdot 0.02040816326530612 \approx 0.0012755102040816326
\][/tex]
Given this value, let's check each provided choice to see if it matches this result.
### Choice A: [tex]\((4 \cdot 7)^{-4}\)[/tex]
Calculate the expression inside the parentheses first:
[tex]\[
4 \cdot 7 = 28
\][/tex]
Now, apply the exponentiation:
[tex]\[
(28)^{-4} = \left(\frac{1}{28}\right)^4 = \frac{1}{28^4}
\][/tex]
This leads us to:
[tex]\[
\frac{1}{28^4} \neq 0.0012755102040816326
\][/tex]
So, Choice A is not equivalent.
### Choice B: [tex]\(\frac{1}{28^2}\)[/tex]
Calculate the expression inside the parentheses first:
[tex]\[
28^2 = 784
\][/tex]
Now, apply the division:
[tex]\[
\frac{1}{28^2} = \frac{1}{784} \approx 0.0012755102040816326
\][/tex]
This matches our earlier result.
So, Choice B is equivalent.
### Choice C: [tex]\(\frac{7^{-2}}{4^2}\)[/tex]
First, evaluate the numerator and the denominator separately:
[tex]\[
7^{-2} = 0.02040816326530612
\][/tex]
[tex]\[
4^2 = 16
\][/tex]
Now, perform the division:
[tex]\[
\frac{7^{-2}}{4^2} = \frac{0.02040816326530612}{16} = 0.0012755102040816326 \div 16 = 0.0012755102040816326
\][/tex]
This value does not match our earlier result.
So, Choice C is not equivalent.
### Choice D: [tex]\((4 \cdot 7)^4\)[/tex]
Calculate the expression inside the parentheses first:
[tex]\[
4 \cdot 7 = 28
\][/tex]
Now, apply the exponentiation:
[tex]\[
(28)^4 = 614656
\][/tex]
Which does not match our earlier result.
So, Choice D is not equivalent.
### Conclusion:
Based on this analysis:
- Choice A: [tex]\((4 \cdot 7)^{-4}\)[/tex] — Not equivalent
- Choice B: [tex]\(\frac{1}{28^2}\)[/tex] — Equivalent
- Choice C: [tex]\(\frac{7^{-2}}{4^2}\)[/tex] — Not equivalent
- Choice D: [tex]\((4 \cdot 7)^4\)[/tex] — Not equivalent
The two correct expressions equivalent to [tex]\(4^{-2} \cdot 7^{-2}\)[/tex] are:
B: [tex]\(\frac{1}{28^2}\)[/tex]
