Sure, let's solve the given expression step-by-step in detail.
We need to evaluate the following expression:
[tex]\[
\frac{3}{8} \text{ of } \left( \frac{7^3}{5} - \frac{1}{3} \left( \frac{11}{4} + 3^{1/3} \right) \times 2^{2/5} \right)
\][/tex]
First, let's break the problem into smaller parts and solve each part individually.
### Step 1: Calculate [tex]\(7^3\)[/tex]
[tex]\[
7^3 = 343
\][/tex]
### Step 2: Calculate [tex]\(\frac{7^3}{5}\)[/tex]
[tex]\[
\frac{7^3}{5} = \frac{343}{5} = 68.6
\][/tex]
### Step 3: Calculate [tex]\(3^{1/3}\)[/tex]
The cube root of 3 will yield approximately:
[tex]\[
3^{1/3} \approx 1.4422
\][/tex]
### Step 4: Calculate [tex]\(\frac{11}{4}\)[/tex]
[tex]\[
\frac{11}{4} = 2.75
\][/tex]
### Step 5: Combine [tex]\(\frac{11}{4} + 3^{1/3}\)[/tex]
[tex]\[
2.75 + 1.4422 \approx 4.1922
\][/tex]
### Step 6: Calculate [tex]\(2^{2/5}\)[/tex]
The fifth root of 2 squared is approximately:
[tex]\[
2^{2/5} \approx 1.3195
\][/tex]
### Step 7: Calculate [tex]\(\frac{1}{3} \left( \frac{11}{4} + 3^{1/3} \right) \times 2^{2/5}\)[/tex]
First, we calculate:
[tex]\[
\frac{1}{3} \times 4.1922 \approx 1.3974
\][/tex]
Then multiply by [tex]\(2^{2/5}\)[/tex]:
[tex]\[
1.3974 \times 1.3195 \approx 1.8439
\][/tex]
### Step 8: Calculate [tex]\(\frac{7^3}{5} - \frac{1}{3}\left( \frac{11}{4} + 3^{1/3} \right) \times 2^{2/5}\)[/tex]
[tex]\[
68.6 - 1.8439 \approx 66.7561
\][/tex]
### Step 9: Calculate [tex]\(\frac{3}{8} \text{ of } \left( \frac{7^3}{5} - \frac{1}{3}\left( \frac{11}{4} + 3^{1/3} \right) \times 2^{2/5} \right)\)[/tex]
[tex]\[
\frac{3}{8} \times 66.7561 \approx 25.0335
\][/tex]
Hence, the final result of the given expression is:
[tex]\[
\boxed{25.0335}
\][/tex]
