To solve the problem, we need to find the numerical values of [tex]\(\sqrt[3]{7}\)[/tex] and [tex]\(\sqrt{7}\)[/tex], then add these values together, and compare the result with the given choices.
First, let's calculate [tex]\(\sqrt[3]{7}\)[/tex]:
[tex]\[
\sqrt[3]{7} \approx 1.912931182772389
\][/tex]
Next, let's calculate [tex]\(\sqrt{7}\)[/tex]:
[tex]\[
\sqrt{7} \approx 2.6457513110645907
\][/tex]
Now, add these two values:
[tex]\[
\sqrt[3]{7} + \sqrt{7} \approx 1.912931182772389 + 2.6457513110645907 = 4.558682493836979
\][/tex]
Next, we compare this result with the provided choices:
(A) [tex]\(\sqrt[4]{7}\)[/tex]:
[tex]\[
\sqrt[4]{7} \approx 1.6265765616977856
\][/tex]
(B) [tex]\(\sqrt[3]{14}\)[/tex]:
[tex]\[
\sqrt[3]{14} \approx 2.4101422641752297
\][/tex]
(C) [tex]\(\sqrt[4]{14}\)[/tex]:
[tex]\[
\sqrt[4]{14} \approx 1.9343364202676694
\][/tex]
(D) [tex]\(\sqrt[3]{49}\)[/tex]:
[tex]\[
\sqrt[3]{49} \approx 3.6593057100229713
\][/tex]
As we can see, the correct answer is the one closest to our calculated result [tex]\(4.558682493836979\)[/tex]. Therefore:
[tex]\[
\boxed{\sqrt[3]{49}}
\][/tex]
