Sure, let's tackle each part of the question step-by-step.
### Part a: Exact Value Calculation
We need to calculate the exact value of the expression:
[tex]\[
\frac{13822 \times 623}{14}
\][/tex]
First, perform the multiplication in the numerator:
[tex]\[
13822 \times 623 = 8,606,806
\][/tex]
Next, divide the result by the denominator, which is 14:
[tex]\[
\frac{8,606,806}{14} = 615,079
\][/tex]
So, the exact value of the expression is:
[tex]\[
\frac{13822 \times 623}{14} = 615,079
\][/tex]
### Part b: Approximation to 1 Significant Figure
To check the accuracy of our exact solution, we'll approximate each number to 1 significant figure and then recompute the expression.
1. Approximate 13822 to 1 significant figure:
[tex]\[
13822 \approx 10000
\][/tex]
2. Approximate 623 to 1 significant figure:
[tex]\[
623 \approx 600
\][/tex]
3. Approximate 14 to 1 significant figure:
[tex]\[
14 \approx 10
\][/tex]
Now, use these approximations in the original expression:
[tex]\[
\frac{10000 \times 600}{10}
\][/tex]
First, perform the multiplication in the numerator:
[tex]\[
10000 \times 600 = 6,000,000
\][/tex]
Next, divide the result by the denominator, which is 10:
[tex]\[
\frac{6,000,000}{10} = 600,000
\][/tex]
So, the approximate value using 1 significant figure is:
[tex]\[
\frac{10000 \times 600}{10} = 600,000
\][/tex]
Comparing this with our exact value:
The exact value is 615,079 and the approximate value is 600,000. These two values are reasonably close to each other, which confirms that our exact calculation is sensible and the order of magnitude is correct.
