Certainly! Let's work through the problem step-by-step and clarify the calculations and results for each part.
### Step-by-step Solution:
#### Given Values:
1. [tex]\( 12 \)[/tex]
2. [tex]\( 4 \)[/tex]
3. [tex]\( 4 \sqrt{2} \)[/tex]
4. [tex]\( 4 \sqrt{3} \)[/tex]
5. [tex]\( 8 \sqrt{2} \)[/tex]
6. [tex]\( 8 \)[/tex]
7. We need to calculate [tex]\( 8 \sqrt{3} \)[/tex]
#### Step 1: List the Given Values
We start by listing all the given values in numerical form, where possible:
[tex]\[ 12, \, 4, \, 4 \sqrt{2}, \, 4 \sqrt{3}, \, 8 \sqrt{2}, \, 8 \][/tex]
We will now calculate the numerical values of [tex]\( 4 \sqrt{2} \)[/tex], [tex]\( 4 \sqrt{3} \)[/tex], and [tex]\( 8 \sqrt{2} \)[/tex].
#### Step 2: Calculate the Values of Expressions involving Square Roots
1. [tex]\( 4 \sqrt{2} \)[/tex]
[tex]\[
4 \sqrt{2} \approx 4 \times 1.41421356237 \approx 5.656854249492381
\][/tex]
2. [tex]\( 4 \sqrt{3} \)[/tex]
[tex]\[
4 \sqrt{3} \approx 4 \times 1.73205080757 \approx 6.928203230275509
\][/tex]
3. [tex]\( 8 \sqrt{2} \)[/tex]
[tex]\[
8 \sqrt{2} \approx 8 \times 1.41421356237 \approx 11.313708498984761
\][/tex]
Now, we include these calculated values in our list:
[tex]\[ 12, \, 4, \, 5.656854249492381, \, 6.928203230275509, \, 11.313708498984761, \, 8 \][/tex]
#### Step 3: Calculate [tex]\( 8 \sqrt{3} \)[/tex]
Finally, we calculate the value of [tex]\( 8 \sqrt{3} \)[/tex]:
1. [tex]\( 8 \sqrt{3} \)[/tex]
[tex]\[
8 \sqrt{3} \approx 8 \times 1.73205080757 \approx 13.856406460551018
\][/tex]
We add this calculated value to our list of numbers:
[tex]\[ 12, \, 4, \, 5.656854249492381, \, 6.928203230275509, \, 11.313708498984761, \, 8, \, 13.856406460551018 \][/tex]
#### Final Result:
Thus, the final list of numbers and the calculated value are:
[tex]\[
\text{Final list of numbers: } [12, 4, 5.656854249492381, 6.928203230275509, 11.313708498984761, 8, 13.856406460551018]
\][/tex]
The specific calculated value of [tex]\( 8 \sqrt{3} \)[/tex] is:
[tex]\[
13.856406460551018
\][/tex]
By going through these steps, we were able to understand and calculate all the necessary values and complete the list correctly.
