The volume of a cube is `778688 mm^(3)`. Find the measure of the edge.

Options

A. 62 mm
B. 72 mm
C. 82 mm
D. 92 mm

Answer

To find the measure of the edge of a cube given its volume, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Volume of a Cube**:
The volume \( V \) of a cube can be expressed as:
\[ V = x^3 \]
where \( x \) is the length of an edge of the cube.

2. **Set Up the Equation**:
We are given that the volume of the cube is \( 778688 \, mm^3 \). Therefore, we can write:
\[ x^3 = 778688 \]

3. **Find the Cube Root**:
To find the edge length \( x \), we need to calculate the cube root of \( 778688 \):
\[ x = \sqrt[3]{778688} \]

4. **Prime Factorization**:
To simplify the calculation, we can perform the prime factorization of \( 778688 \):
- Divide by \( 2 \):
\[ 778688 \div 2 = 389344 \]
- Divide by \( 2 \) again:
\[ 389344 \div 2 = 194672 \]
- Continue dividing by \( 2 \):
\[ 194672 \div 2 = 97336 \]
\[ 97336 \div 2 = 48668 \]
\[ 48668 \div 2 = 24334 \]
\[ 24334 \div 2 = 12167 \]
- Now, \( 12167 \) is not divisible by \( 2 \), so we try \( 23 \):
\[ 12167 \div 23 = 529 \]
- Finally, \( 529 \) can be factored as:
\[ 529 = 23 \times 23 \]

5. **Combine the Factors**:
The prime factorization of \( 778688 \) is:
\[ 778688 = 2^6 \times 23^3 \]

6. **Calculate the Cube Root**:
To find \( x \), we take the cube root of the prime factorization:
\[ x = \sqrt[3]{2^6 \times 23^3} = \sqrt[3]{(2^2)^3 \times (23^3)} = 2^2 \times 23 = 4 \times 23 = 92 \]

7. **Conclusion**:
The measure of the edge of the cube is:
\[ x = 92 \, mm \]

### Final Answer:
The measure of the edge of the cube is \( 92 \, mm \).