Let's simplify each expression to determine the product or quotient in scientific notation.
1. Expression: [tex]\((3.8 \times 10^3) \cdot (9.4 \times 10^{-5})\)[/tex]
[tex]\[3.8 \times 9.4 = 35.72\][/tex]
[tex]\[10^3 \times 10^{-5} = 10^{-2}\][/tex]
So, [tex]\(35.72 \times 10^{-2} = 3.572 \times 10^{-1}\)[/tex]
Rounded to the tenths place: [tex]\(0.3572\)[/tex]
2. Expression: [tex]\((4.2 \times 10^7) \cdot (7.4 \times 10^{-2})\)[/tex]
[tex]\[4.2 \times 7.4 = 31.08\][/tex]
[tex]\[10^7 \times 10^{-2} = 10^5\][/tex]
So, [tex]\(31.08 \times 10^5 = 3.108 \times 10^6\)[/tex]
Rounded to the tenths place: [tex]\(3.1 \times 10^6\)[/tex]
3. Expression: [tex]\(\frac{(8.6 \times 10^{-6}) \cdot (7.1 \times 10^{-6})}{(4.1 \times 10^2) \cdot (2.8 \times 10^{-7})}\)[/tex]
[tex]\[8.6 \times 7.1 = 61.06\][/tex]
[tex]\[10^{-6} \times 10^{-6} = 10^{-12}\][/tex]
[tex]\[4.1 \times 2.8 = 11.48\][/tex]
[tex]\[10^2 \times 10^{-7} = 10^{-5}\][/tex]
[tex]\[\frac{61.06 \times 10^{-12}}{11.48 \times 10^{-5}} = \frac{61.06}{11.48} \times 10^{-7} = 5.32 \times 10^{-7}\][/tex]
Rounded to the tenths place: [tex]\(5.3 \times 10^{-6}\)[/tex]
4. Expression: [tex]\(\frac{(6.9 \times 10^{-4}) \cdot (7.7 \times 10^{-6})}{(2.7 \times 10^{-2}) \cdot (4.7 \times 10^{-7})}\)[/tex]
[tex]\[6.9 \times 7.7 = 53.13\][/tex]
[tex]\[10^{-4} \times 10^{-6} = 10^{-10}\][/tex]
[tex]\[2.7 \times 4.7 = 12.69\][/tex]
[tex]\[10^{-2} \times 10^{-7} = 10^{-5}\][/tex]
[tex]\[\frac{53.13 \times 10^{-10}}{12.69 \times 10^{-5}} = \frac{53.13}{12.69} \times 10^{-5} \times 10^{-5} = 4.19 \times 10^{-15}\][/tex]
Rounded to the tenths place: [tex]\(4.2 \times 10^{-15}\)[/tex]
Now, match each simplified expression with the given options:
- [tex]\(\left(3.8 \times 10^3\right) \cdot\left(9.4 \times 10^{-5}\right) \rightarrow 0.3572\)[/tex]
- [tex]\(\left(4.2 \times 10^7\right) \cdot\left(7.4 \times 10^{-2}\right) \rightarrow 3.1 \times 10^6\)[/tex]
- [tex]\(\frac{\left(8.6 \times 10^{-6}\right) \cdot\left(7.1 \times 10^{-6}\right)}{\left(4.1 \times 10^2\right) \cdot\left(2.8 \times 10^{-7}\right)} \rightarrow 5.3 \times 10^{-6}\)[/tex]
- [tex]\(\frac{\left(6.9 \times 10^{-4}\right) \cdot\left(7.7 \times 10^{-6}\right)}{\left(2.7 \times 10^{-2}\right) \cdot\left(4.7 \times 10^{-7}\right)} \rightarrow 4.2 \times 10^{-15}\)[/tex]
So, the correct pairs are:
- [tex]\(\left(3.8 \times 10^3\right) \cdot\left(9.4 \times 10^{-5}\right)\)[/tex] – [tex]$3.6 \times 10^{-1}$[/tex]
- [tex]\(\left(4.2 \times 10^7\right) \cdot\left(7.4 \times 10^{-2}\right)\)[/tex] – [tex]$3.1 \times 10^6$[/tex]
- [tex]\(\frac{\left(8.6 \times 10^{-6}\right) \cdot\left(7.1 \times 10^{-6}\right)}{\left(4.1 \times 10^2\right) \cdot\left(2.8 \times 10^{-7}\right)}\)[/tex] – [tex]$5.3 \times 10^{-6}$[/tex]
- [tex]\(\frac{\left(6.9 \times 10^{-4}\right) \cdot\left(7.7 \times 10^{-6}\right)}{\left(2.7 \times 10^{-2}\right) \cdot\left(4.7 \times 10^{-7}\right)}\)[/tex] – [tex]$4.2 \times 10^{-15}$[/tex]
