The given expression involves cube roots:

[ \sqrt[3]{81} - \sqrt[3]{24} ]

Step 1: Approximate the cube roots

1. $\sqrt[3]{81}$:
   We know $4^3 = 64$ and $5^3 = 125$. Since 81 lies between these two, we estimate: $\sqrt[3]{81} \approx 4.3$

2. $\sqrt[3]{24}$:
   Similarly, $2^3 = 8$ and $3^3 = 27$. Since 24 lies between these values: $\sqrt[3]{24} \approx 2.884$

Step 2: Subtract the two values:

$4.3 - 2.884 \approx 1.416$

Thus, the approximate result is $1.416$.

Would you like a detailed exact form explanation or further clarification?

Here are 5 related questions you could explore:

1. What are the properties of cube roots for irrational numbers?
2. How do cube roots differ from square roots in behavior?
3. Can you simplify $\sqrt[3]{81} - \sqrt[3]{24}$ further?
4. How is estimating cube roots helpful for larger values?
5. What are the methods for manually approximating cube roots?

Tip: Understanding perfect cubes around your number helps in quick estimations.