Let's solve the last two expressions, $(fg)(5)$ and $\left(\frac{f}{g}\right)(5)$, given the functions:

$f(x) = 4x^4 \quad \text{and} \quad g(x) = 24x^{1/3}.$

1. Compute $f(5)$ and $g(5)$:

   • $f(5) = 4 \cdot 5^4 = 4 \cdot 625 = 2500$.

   • $g(5) = 24 \cdot 5^{1/3}$.

     To approximate $5^{1/3}$, we get about $1.71$, so:

     $g(5) \approx 24 \cdot 1.71 = 41.04.$

2. Calculate $(fg)(5)$:

   $(fg)(5) = f(5) \cdot g(5) = 2500 \cdot 41.04 \approx 102600.$

3. Calculate $\left(\frac{f}{g}\right)(5)$:

   $\left(\frac{f}{g}\right)(5) = \frac{f(5)}{g(5)} = \frac{2500}{41.04} \approx 60.91.$

Final answers:

• $(fg)(5) \approx 102600$.
• $\left(\frac{f}{g}\right)(5) \approx 60.91$.

Would you like further explanation on any steps?

Follow-up Questions:

1. How is the product of two functions calculated?
2. What does the notation $\left(\frac{f}{g}\right)(x)$ represent?
3. Why is it important to evaluate each function separately before combining them?
4. How does rounding affect answers in calculations like these?
5. How can we approximate roots like $5^{1/3}$ accurately?

Tip: When solving function compositions, break down each function evaluation separately for clarity.