Question 35
Prove that the sum $$
\sqrt[3]{\frac{a+1}{2}+\frac{a+3}{6} \sqrt{\frac{4 a+3}{3}}}+\sqrt[3]{\frac{a+1}{2}-\frac{a+3}{6} \sqrt{\frac{4 a+3}{3}}}
$$ is independent of the value of $a$, for all real value $a \geq -3/4$, and find the value of that sum.
\sqrt[3]{\frac{a+1}{2}+\frac{a+3}{6} \sqrt{\frac{4 a+3}{3}}}+\sqrt[3]{\frac{a+1}{2}-\frac{a+3}{6} \sqrt{\frac{4 a+3}{3}}}
$$ is independent of the value of $a$, for all real value $a \geq -3/4$, and find the value of that sum.