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]
