To solve this problem, we need to find the smallest number by which 8232 must be divided to obtain a perfect cube, and then find the cube root of the resulting number.

Step 1: Prime Factorization of 8232

We begin by performing the prime factorization of 8232:

$8232 \div 2 = 4116$ $4116 \div 2 = 2058$ $2058 \div 2 = 1029$ $1029 \div 3 = 343$ $343 = 7^3$

Thus, the prime factorization of 8232 is: $8232 = 2^3 \times 3 \times 7^3$

Step 2: Identify Missing Factors to Make a Perfect Cube

For a number to be a perfect cube, the exponents of all its prime factors must be multiples of 3. In the prime factorization of 8232, we see:

• $2^3$ is already a cube.
• $3^1$ is not a cube (it needs two more factors of 3).
• $7^3$ is already a cube.

So, we need to divide by $3$ (to remove the extra factor of 3) to make it a perfect cube.

Step 3: Divide 8232 by 3

$\frac{8232}{3} = 2744$

Step 4: Find the Cube Root of 2744

Now, we find the cube root of 2744: $\sqrt[3]{2744} = 14$

Conclusion:

• The least number by which 8232 must be divided to obtain a perfect cube is 3.
• The cube root of the resulting number (2744) is 14.

Would you like further clarification or additional details?

Here are 5 related questions to deepen your understanding:

1. How is the prime factorization method useful in determining perfect cubes?
2. What happens if a number's prime factors all have exponents that are multiples of 3?
3. How would you find the least number to multiply by, instead of divide, to get a perfect cube?
4. What other methods can be used to check if a number is a perfect cube?
5. How can you apply this method to much larger numbers?

Tip: When checking for perfect cubes, always ensure that the exponents in the prime factorization are divisible by 3.