The side of solid metallic cube is 50 cm. The cube is melted and recast into 8000 equal solid cubical dice. Determine the side of the dice.

Answer

To solve the problem step by step, we will find the side of the small cubical dice after melting the original cube.

### Step 1: Calculate the volume of the original cube
The formula for the volume \( V \) of a cube with side length \( a \) is given by:
\[ V = a^3 \]
Given that the side of the original cube is 50 cm, we can calculate its volume:
\[ V = 50^3 = 50 \times 50 \times 50 = 125000 \, \text{cm}^3 \]

### Step 2: Set up the equation for the volume of the small cubes
When the original cube is melted and recast into 8000 smaller cubes (dice), the total volume of the smaller cubes will be equal to the volume of the original cube. Let the side of each small cube be \( x \). The volume of one small cube is:
\[ V_{\text{small}} = x^3 \]
Since there are 8000 such small cubes, the total volume of the small cubes is:
\[ V_{\text{total small}} = 8000 \times x^3 \]

### Step 3: Equate the volumes
Since the volume of the original cube is equal to the total volume of the small cubes, we have:
\[ 125000 = 8000 \times x^3 \]

### Step 4: Solve for \( x^3 \)
To find \( x^3 \), we rearrange the equation:
\[ x^3 = \frac{125000}{8000} \]
Calculating the right-hand side:
\[ x^3 = \frac{125000 \div 1000}{8000 \div 1000} = \frac{125}{8} \]

### Step 5: Find \( x \)
Now, we need to find \( x \) by taking the cube root of both sides:
\[ x = \sqrt[3]{\frac{125}{8}} = \frac{\sqrt[3]{125}}{\sqrt[3]{8}} = \frac{5}{2} = 2.5 \, \text{cm} \]

### Final Answer
The side of each small cubical die is:
\[ \text{Side of the dice} = 2.5 \, \text{cm} \]

---