Three cubes whose edges are x cm, 8 cm, and 10 cm respectively, are melted and recast into a single cube of edge 12 cm. Find 'x'.

Answer

To solve the problem, we need to find the value of \( x \) given that three cubes with edges \( x \) cm, \( 8 \) cm, and \( 10 \) cm are melted and recast into a single cube with an edge of \( 12 \) cm.

### Step-by-Step Solution:

1. **Calculate the Volume of Each Cube:**
- The volume of a cube is given by the formula:
\[ \text{Volume} = \text{side}^3 \]
- For the first cube with edge \( x \):
\[ V_1 = x^3 \]
- For the second cube with edge \( 8 \) cm:
\[ V_2 = 8^3 = 512 \text{ cm}^3 \]
- For the third cube with edge \( 10 \) cm:
\[ V_3 = 10^3 = 1000 \text{ cm}^3 \]

2. **Calculate the Volume of the New Cube:**
- The volume of the new cube with edge \( 12 \) cm is:
\[ V_{\text{new}} = 12^3 = 1728 \text{ cm}^3 \]

3. **Set Up the Equation:**
- According to the problem, the total volume of the three smaller cubes equals the volume of the new cube:
\[ V_1 + V_2 + V_3 = V_{\text{new}} \]
- Substituting the volumes we calculated:
\[ x^3 + 512 + 1000 = 1728 \]

4. **Simplify the Equation:**
- Combine the known volumes:
\[ x^3 + 1512 = 1728 \]

5. **Isolate \( x^3 \):**
- Subtract \( 1512 \) from both sides:
\[ x^3 = 1728 - 1512 \]
\[ x^3 = 216 \]

6. **Find \( x \):**
- To find \( x \), take the cube root of both sides:
\[ x = \sqrt[3]{216} \]
- Since \( 216 = 6^3 \):
\[ x = 6 \text{ cm} \]

### Final Answer:
The value of \( x \) is \( 6 \) cm.

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