ABC26GN1277 · Simplification

$$\begin{aligned} \left(2 p+\frac{1}{p}\right)=4 & \Rightarrow \frac{1}{2}\left(2 p+\frac{1}{p}\right)=\frac{1}{2} \times 4 \Rightarrow\left(p+\frac{1}{2 p}\right)=2 \\ & \Rightarrow\left(p+\frac{1}{2 p}\right)^{3}=2^{3}=8 \\ & \Rightarrow p^{3}+\frac{1}{8 p^{3}}+3 p \cdot \frac{1}{2 p}\left(p+\frac{1}{2 p}\right)=8 \\ & \Rightarrow p^{3}+\frac{1}{8 p^{3}}+3 \times \frac{1}{2} \times 2=8 \\ & \Rightarrow p^{3}+\frac{1}{8 p^{3}}=8-3=5 \end{aligned}$$