Let's break down and simplify each part of the given expressions step by step.
### First Part:
Simplify [tex]\(\frac{1}{2}\)[/tex] of [tex]\(\frac{31}{2} + \frac{11}{2} \left(2^{1/2} - \frac{2}{3}\right)\)[/tex]
1. Calculate the expression inside the parentheses first:
[tex]\[
2^{1/2} \approx 1.414213562 \quad \text{(the square root of 2)}
\][/tex]
Therefore,
[tex]\[
2^{1/2} - \frac{2}{3} \approx 1.414213562 - 0.666666667 \approx 0.747546895
\][/tex]
2. Multiply [tex]\(\frac{11}{2}\)[/tex] by [tex]\((2^{1/2} - \frac{2}{3})\)[/tex]:
[tex]\[
\frac{11}{2} \times 0.747546895 \approx 11 \times 0.373773447 \approx 4.111508917
\][/tex]
3. Add [tex]\(\frac{31}{2}\)[/tex] to the result obtained in step 2:
[tex]\[
\frac{31}{2} \approx 15.5
\][/tex]
Therefore,
[tex]\[
15.5 + 4.111508917 \approx 19.611508917
\][/tex]
4. Now, take [tex]\(\frac{1}{2}\)[/tex] of the result:
[tex]\[
\frac{1}{2} \times 19.611508917 \approx 9.805754459
\][/tex]
### Second Part:
Simplify [tex]\(\frac{3}{4}\)[/tex] of [tex]\(2^{1/2} \div \frac{1}{2}\)[/tex]
1. Calculate [tex]\(2^{1/2}\)[/tex] again (already done above):
[tex]\[
2^{1/2} \approx 1.414213562
\][/tex]
2. Divide [tex]\(2^{1/2}\)[/tex] by [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
1.414213562 \div \frac{1}{2} = 1.414213562 \times 2 \approx 2.828427124
\][/tex]
3. Now, take [tex]\(\frac{3}{4}\)[/tex] of the result:
[tex]\[
\frac{3}{4} \times 2.828427124 \approx \left(\frac{3 \times 2.828427124}{4}\right) \approx 2.121320343
\][/tex]
### Final Results:
Putting it all together, we get:
- The simplified result for the first part:
[tex]\[
9.805753963192679
\][/tex]
- The simplified result for the second part:
[tex]\[
2.121320343559643
\][/tex]
Thus, the detailed, step-by-step solutions yield the answers:
[tex]\[
9.805753963192679 \quad \text{and} \quad 2.121320343559643
\][/tex]
