Question:

Step 1: Set up substitution and constraint Let $t = a + \frac{1}{a}$. For $a > 0$, by the AM-GM inequality, we have $a + \frac{1}{a} \ge 2\sqrt{a \cdot \frac{1}{a}} = 2$. Thus, $t \ge 2$. Step 2: Express $a^2 + \frac{1}{a^2}$ in terms of $t$ Squaring the substitution, we get: $$t^2 = \left(a + \frac{1}{a}\right)^2 = a^2 + 2\left(a \cdot \frac{1}{a}\right) + \frac{1}{a^2} = a^2 + 2 + \frac{1}{a^2}$$ Rearranging, we find: $$a^2 + \frac{1}{a^2} = t^2 - 2$$ Step 3: Express $a^4 + \frac{1}{a^4}$ in terms of $t$ Squaring the expression for $a^2 + \frac{1}{a^2}$: $$\left(a^2 + \frac{1}{a^2}\right)^2 = a^4 + 2\left(a^2 \cdot \frac{1}{a^2}\right) + \frac{1}{a^4} = a^4 + 2 + \frac{1}{a^4}$$ Substituting $a^2 + \frac{1}{a^2} = t^2 - 2$: $$a^4 + \frac{1}{a^4} = (t^2 - 2)^2 - 2 = (t^4 - 4t^2 + 4) - 2 = t^4 - 4t^2 + 2$$ Step 4: Simplify the numerator of the expression The numerator is $\left(a + \frac{1}{a}\right)^4 - \left(a^4 + \frac{1}{a^4}\right) - 2$. Substituting the expressions in terms of $t$: $$\text{Numerator} = t^4 - (t^4 - 4t^2 + 2) - 2$$ $$= t^4 - t^4 + 4t^2 - 2 - 2$$ $$= 4t^2 - 4 = 4(t^2 - 1)$$ Step 5: Simplify the denominator of the expression The denominator is $\left(a + \frac{1}{a}\right)^2 + a^2 + \frac{1}{a^2}$. Substituting the expressions in terms of $t$: $$\text{Denominator} = t^2 + (t^2 - 2)$$ $$= 2t^2 - 2 = 2(t^2 - 1)$$ Step 6: Evaluate the expression and determine its minimum value The given expression is the ratio of the simplified numerator and denominator: $$\frac{4(t^2 - 1)}{2(t^2 - 1)}$$ Since $t \ge 2$, it follows that $t^2 \ge 4$. Therefore, $t^2 - 1 \ge 3$. Since $t^2 - 1$ is never zero, we can cancel the term $(t^2 - 1)$ from the numerator and denominator. The expression simplifies to: $$\frac{4}{2} = 2$$ Since the expression evaluates to a constant value of 2 for all $a > 0$, its minimum value is 2.