Sure, let's solve the problems step-by-step and express the answers to the correct number of significant figures.
### Problem 1: Division
We need to calculate the division of [tex]\(2.31\)[/tex] by [tex]\(0.790\)[/tex]:
[tex]\[ \frac{2.31}{0.790} \][/tex]
To find this, we simply perform the division:
[tex]\[ 2.31 \div 0.790 \][/tex]
Since both numbers have 3 significant figures, the result should also be rounded to 3 significant figures. Performing the division:
[tex]\[ 2.31 \div 0.790 \approx 2.924 \][/tex]
Rounding this to 3 significant figures, the answer is:
[tex]\[ \boxed{2.92} \][/tex]
### Problem 2: Multiplication with Scientific Notation
We need to calculate the product of [tex]\(2.08 \times 10^3\)[/tex] and [tex]\(3.11 \times 10^2\)[/tex]:
[tex]\[ \left(2.08 \times 10^3\right) \times \left(3.11 \times 10^2\right) \][/tex]
First, multiply the base numbers:
[tex]\[ 2.08 \times 3.11 = 6.4648 \][/tex]
Now, combine the powers of 10:
[tex]\[ 10^3 \times 10^2 = 10^{3+2} = 10^5 \][/tex]
Thus, the product is:
[tex]\[ 6.4648 \times 10^5 \][/tex]
Both initial numbers, [tex]\(2.08\)[/tex] and [tex]\(3.11\)[/tex], have 3 significant figures, so the final result should also be rounded to 3 significant figures:
[tex]\[ 6.4648 \approx 6.47 \][/tex]
Thus, multiplying by [tex]\(10^5\)[/tex], we get:
[tex]\[ \boxed{6.4688} \times 10^5 \][/tex]
So, the answers to the problems are:
[tex]\[
\frac{2.31}{0.790} = \boxed{2.92}
\][/tex]
[tex]\[
\left(2.08 \times 10^3\right) \times \left(3.11 \times 10^2\right) = \boxed{6.4688} \times 10^5
\][/tex]
