The given expression is:

[ 7 \cdot \sqrt[3]{54} + 2 \cdot \sqrt[3]{128} ]

We need to simplify each radical and then combine the like terms.

Step 1: Simplify $\sqrt[3]{54}$

The number 54 can be factored as: $54 = 2 \times 3^3$ So, we can simplify $\sqrt[3]{54}$ as: [ \sqrt[3]{54} = \sqrt[3]{2 \times 3^3} = 3 \cdot \sqrt[3]{2} ]

Now, the first term becomes: [ 7 \cdot 3 \cdot \sqrt[3]{2} = 21 \cdot \sqrt[3]{2} ]

Step 2: Simplify $\sqrt[3]{128}$

The number 128 can be factored as: $128 = 2^7$ So, we can simplify $\sqrt[3]{128}$ as: [ \sqrt[3]{128} = \sqrt[3]{2^7} = 2^2 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} ]

Now, the second term becomes: [ 2 \cdot 4 \cdot \sqrt[3]{2} = 8 \cdot \sqrt[3]{2} ]

Step 3: Combine like terms

Both terms involve $\sqrt[3]{2}$, so we can combine them: [ 21 \cdot \sqrt[3]{2} + 8 \cdot \sqrt[3]{2} = (21 + 8) \cdot \sqrt[3]{2} = 29 \cdot \sqrt[3]{2} ]

Final Answer:

$29 \cdot \sqrt[3]{2}$

Would you like more details or have any questions? Here are some related problems:

1. Simplify $5 \cdot \sqrt[3]{16} + 3 \cdot \sqrt[3]{32}$.
2. Find the value of $2 \cdot \sqrt[3]{250} + 4 \cdot \sqrt[3]{500}$.
3. Simplify $\sqrt[3]{54} + \sqrt[3]{250}$.
4. Combine $9 \cdot \sqrt[3]{27} + 3 \cdot \sqrt[3]{81}$.
5. Simplify $6 \cdot \sqrt[3]{500} + 4 \cdot \sqrt[3]{64}$.

Tip: When simplifying cube roots, look for the largest perfect cube factor of the number under the radical!