The edges of three solid cubes are 6cm , 8cm and 10 cm These cubes are melted and recasted into a single cube. Find the edge of the resulting cube.

Answer

To solve the problem of finding the edge of the resulting cube formed by melting three cubes with edges of 6 cm, 8 cm, and 10 cm, we will follow these steps:

### Step 1: Calculate the volume of each cube.
The volume \( V \) of a cube is given by the formula:
\[ V = \text{side}^3 \]

1. **Volume of the first cube (edge = 6 cm)**:
\[ V_1 = 6^3 = 6 \times 6 \times 6 = 216 \, \text{cm}^3 \]

2. **Volume of the second cube (edge = 8 cm)**:
\[ V_2 = 8^3 = 8 \times 8 \times 8 = 512 \, \text{cm}^3 \]

3. **Volume of the third cube (edge = 10 cm)**:
\[ V_3 = 10^3 = 10 \times 10 \times 10 = 1000 \, \text{cm}^3 \]

### Step 2: Calculate the total volume of the three cubes.
Now, we add the volumes of the three cubes:
\[ \text{Total Volume} = V_1 + V_2 + V_3 = 216 + 512 + 1000 \]
Calculating the total:
\[ 216 + 512 = 728 \]
\[ 728 + 1000 = 1728 \, \text{cm}^3 \]

### Step 3: Find the edge of the new cube.
Let the edge of the new cube be \( a \). The volume of the new cube is given by:
\[ V = a^3 \]
Since the total volume from the melted cubes is 1728 cm³, we have:
\[ a^3 = 1728 \]

### Step 4: Calculate the cube root to find the edge length.
To find \( a \), we need to calculate the cube root of 1728:
\[ a = \sqrt[3]{1728} \]

Using prime factorization:
\[ 1728 = 2^6 \times 3^3 \]
Taking the cube root:
\[ a = \sqrt[3]{2^6} \times \sqrt[3]{3^3} = 2^{6/3} \times 3^{3/3} = 2^2 \times 3^1 = 4 \times 3 = 12 \, \text{cm} \]

### Final Answer:
The edge of the resulting cube is **12 cm**.

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