To find the volume of the largest cube that can be cut from the rectangular block, follow these steps:

1. Determine the largest possible cube edge length:

The edge length of the cube must fit within the dimensions of the rectangular block. Therefore, the edge length of the cube cannot exceed the smallest dimension of the block.

The dimensions of the rectangular block are: $15 \, \text{cm}, \, 12 \, \text{cm}, \, 18 \, \text{cm}.$ The smallest dimension is $12 \, \text{cm}$. Thus, the largest cube that can be cut has an edge length of $12 \, \text{cm}$.

2. Calculate the volume of the cube:

The volume $V$ of a cube is given by: $V = \text{edge length}^3$ Substitute $\text{edge length} = 12 \, \text{cm}$: $V = 12^3 = 12 \times 12 \times 12 = 1728 \, \text{cm}^3.$

Final Answer:

The volume of the largest cube that can be cut from the rectangular block is: $\boxed{1728 \, \text{cm}^3}.$

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Would you like further explanation or related examples? Here are 5 related questions for practice:

1. What is the total volume of the rectangular block?
2. How many such cubes of edge length 12 cm can be cut from the block?
3. What would be the largest cube if the smallest dimension were reduced to 10 cm?
4. How much volume of the block would remain after cutting out the cube?
5. Can multiple smaller cubes be cut from the remaining block after removing the largest cube?

Tip: When solving problems involving fitting shapes, always compare dimensions to identify the limiting factor.