1. Problem: Given $p + \frac{1}{p} = 5$, find the value of $p^3 + \frac{1}{p^3}$. 2. Start with the identity: $$\left(p + \frac{1}{p}\right)^3 = p^3 + \frac{1}{p^3} + 3\left(p + \frac{1}{p}\right)$$ 3. Substitute $p + \frac{1}{p} = 5$ into the equation: $$5^3 = p^3 + \frac{1}{p^3} + 3 \times 5$$ 4. Simplify the left side: $$125 = p^3 + \frac{1}{p^3} + 15$$ 5. Solve for $p^3 + \frac{1}{p^3}$: $$p^3 + \frac{1}{p^3} = 125 - 15 = 110$$ Final answer: $110$