Let's solve each portion of the question step by step.
Part (a): Calculate [tex]\((5)^{3 / 8}\)[/tex]
To calculate [tex]\( (5)^{\frac{3}{8}} \)[/tex], we need to understand that the exponent [tex]\(\frac{3}{8}\)[/tex] means we first take the 8th root of 5 and then raise the result to the power of 3. Alternatively, it can be interpreted as raising 5 to the power of [tex]\(\frac{3}{8}\)[/tex] directly. The result for [tex]\((5)^{3 / 8}\)[/tex] is approximately [tex]\(1.8285790999795744\)[/tex].
Part (b): Calculate [tex]\(\left(\frac{5}{2}\right)^{1 / 2}\)[/tex]
To calculate [tex]\(\left(\frac{5}{2}\right)^{\frac{1}{2}}\)[/tex], we need to find the square root of [tex]\(\frac{5}{2}\)[/tex]. The square root function is the same as raising the number to the power of [tex]\(\frac{1}{2}\)[/tex]. Therefore, [tex]\(\left(\frac{5}{2}\right)^{\frac{1}{2}}\)[/tex] approximates to [tex]\(1.5811388300841898\)[/tex].
Part (c): Calculate [tex]\(\left(\frac{4}{7}\right)^{7 / 2}\)[/tex]
For this part, we need to compute [tex]\(\left(\frac{4}{7}\right)^{\frac{7}{2}}\)[/tex]. This can be understood as raising [tex]\(\frac{4}{7}\)[/tex] to the power of [tex]\(\frac{7}{2}\)[/tex]. The calculation results in approximately [tex]\(0.14104796660402646\)[/tex].
So the solutions to the given expressions are:
(a) [tex]\((5)^{3 / 8} \approx 1.8285790999795744\)[/tex]
(b) [tex]\(\left(\frac{5}{2}\right)^{1 / 2} \approx 1.5811388300841898\)[/tex]
(c) [tex]\(\left(\frac{4}{7}\right)^{7 / 2} \approx 0.14104796660402646\)[/tex]
