Sure, let's break this down step by step:
First, let's evaluate the numerator:
[tex]\[
\left(1^{\frac{3}{7}} - \frac{5}{8}\right) \times 2^{-1}
\][/tex]
We know that any number raised to any power of 1 is just 1, so:
[tex]\[
1^{\frac{3}{7}} = 1
\][/tex]
Now substitute back into the numerator expression:
[tex]\[
\left(1 - \frac{5}{8}\right) \times 2^{-1}
\][/tex]
Now calculate [tex]\(1 - \frac{5}{8}\)[/tex]:
[tex]\[
1 - \frac{5}{8} = \frac{8}{8} - \frac{5}{8} = \frac{3}{8}
\][/tex]
Next, we need to calculate [tex]\(\frac{3}{8} \times 2^{-1}\)[/tex]. [tex]\(2^{-1}\)[/tex] is the same as [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
\frac{3}{8} \times \frac{1}{2} = \frac{3}{16}
\][/tex]
So, the numerator evaluates to [tex]\(\frac{3}{16}\)[/tex].
Now let's move on to the denominator:
[tex]\[
\frac{3}{4} + 1^{\frac{5}{7}} \div 4 \times 2^{\frac{1}{3}}
\][/tex]
As before, [tex]\(1^{\frac{5}{7}} = 1\)[/tex].
So now the expression becomes:
[tex]\[
\frac{3}{4} + \frac{1}{4} \times 2^{\frac{1}{3}}
\][/tex]
Next, let's calculate [tex]\(2^{\frac{1}{3}}\)[/tex]. This is the cube root of 2, approximately [tex]\(1.25992\)[/tex].
Now multiply this value by [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
\frac{1}{4} \times 1.25992 \approx 0.31498
\][/tex]
Add this result to [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[
\frac{3}{4} + 0.31498 = 0.75 + 0.31498 \approx 1.06498
\][/tex]
So, the denominator evaluates to approximately [tex]\(1.06498\)[/tex].
Finally, we need to divide the numerator by the denominator:
[tex]\[
\text{Numerator} = \frac{3}{16} \approx 0.1875
\][/tex]
[tex]\[
\text{Denominator} \approx 1.06498
\][/tex]
So:
[tex]\[
\frac{0.1875}{1.06498} \approx 0.17606
\][/tex]
Therefore, the evaluated expression is approximately [tex]\(0.17606\)[/tex].
