Answer :

Certainly! Let's evaluate the given expression step by step:

[tex]\[ \frac{\left(1^{3/7} - \frac{5}{8}\right) \times \frac{2}{3}}{\frac{3}{4} + \left(1^{5/7} \div 7\right) \times 2^{1/3}} \][/tex]

### Step 1: Simplify the Numerator

First, we'll evaluate each component in the numerator.

1. Evaluate [tex]\(1^{3/7}\)[/tex]:
[tex]\[ 1^{3/7} = 1.0 \][/tex]

2. Evaluate [tex]\(\frac{5}{8}\)[/tex]:
[tex]\[ \frac{5}{8} = 0.625 \][/tex]

3. Subtract the two results:
[tex]\[ 1.0 - 0.625 = 0.375 \][/tex]

4. Multiply by [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[ 0.375 \times \frac{2}{3} = 0.25 \][/tex]

So, the numerator simplifies to:
[tex]\[ 0.25 \][/tex]

### Step 2: Simplify the Denominator

Next, we’ll evaluate each component in the denominator.

1. Evaluate [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[ \frac{3}{4} = 0.75 \][/tex]

2. Evaluate [tex]\(1^{5/7}\)[/tex]:
[tex]\[ 1^{5/7} = 1.0 \][/tex]

3. Evaluate [tex]\(2^{1/3}\)[/tex]:
[tex]\[ 2^{1/3} \approx 1.2599210498948732 \][/tex]

4. Divide [tex]\(1^{5/7}\)[/tex] by 7:
[tex]\[ \frac{1.0}{7} = 0.14285714285714285 \][/tex]

5. Multiply the result by [tex]\(2^{1/3}\)[/tex]:
[tex]\[ 0.14285714285714285 \times 1.2599210498948732 \approx 0.17998872141355332 \][/tex]

6. Add [tex]\(\frac{3}{4}\)[/tex] and the result:
[tex]\[ 0.75 + 0.17998872141355332 \approx 0.9299887214135533 \][/tex]

So, the denominator simplifies to:
[tex]\[ 0.9299887214135533 \][/tex]

### Step 3: Divide the Numerator by the Denominator

Finally, we divide the simplified numerator by the simplified denominator:

[tex]\[ \frac{0.25}{0.9299887214135533} \approx 0.2688204644245663 \][/tex]

The result of the given expression is:

[tex]\[ \boxed{0.2688204644245663} \][/tex]