The cubes if edges 6,8,10cms are melted without loss of metal into a single cube.the edge of the new cuble will be 

To find the edge length of the new cube formed by melting three cubes with edges of 6 cm, 8 cm, and 10 cm, we follow these steps:

1. Calculate the volumes of the original cubes:
   \[
   V_1 = 6^3 = 216 \text{ cm}^3
   \]
   \[
   V_2 = 8^3 = 512 \text{ cm}^3
   \]
   \[
   V_3 = 10^3 = 1000 \text{ cm}^3
   \]

2. Sum the volumes to get the total volume:
   \[
   V_{\text{total}} = V_1 + V_2 + V_3 = 216 + 512 + 1000 = 1728 \text{ cm}^3
   \]

3. To find the edge length of the new cube, we take the cube root of the total volume:
   \[
   a = \sqrt[3]{1728}
   \]

4. Since \(1728\) is a perfect cube:
   \[
   a = 12 \text{ cm}
   \]

Therefore, the edge length of the new cube is 12 cm.

Would you like more details or have any questions about the solution? Here are some related questions:

1. What is the volume of a cube with an edge length of 12 cm?
2. How do you calculate the surface area of a cube with an edge length of 12 cm?
3. If you melt a cube of edge 5 cm and a cube of edge 7 cm, what will be the edge length of the resulting cube?
4. How do you find the diagonal of a cube with edge length 12 cm?
5. What is the formula for the surface area of a cube?
6. If you triple the edge length of a cube, by what factor does the volume increase?
7. What is the difference between the volume of a cube with edge length 9 cm and a cube with edge length 12 cm?
8. How do changes in edge length affect the surface area of a cube?

**Tip:** When combining volumes of geometric shapes, always ensure the total volume is considered correctly before applying the geometric properties of the resulting shape.

Learn how to calculate the edge length of a new cube formed by melting cubes with edges of 6 cm, 8 cm, and 10 cm. Follow a step-by-step solution involving volume calculation and cube root. Suitable for students in Grades 7-9.

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