To solve this problem, we need to find the smallest multiple of 3, 4, and 6 that is a perfect cube, and then verify if its cube root is divisible by 3, 4, and 6.

Step 1: Prime Factorization of 3, 4, and 6

• $3 = 3^1$
• $4 = 2^2$
• $6 = 2^1 \times 3^1$

To find a common multiple, we'll take the highest powers of all prime factors involved, which are $2^2$ (from 4) and $3^1$ (from 3 and 6). However, we also want the number to be a perfect cube. For a number to be a perfect cube, all exponents in its prime factorization must be divisible by 3.

Step 2: Make the number a perfect cube

Currently, the prime factorization is $2^2 \times 3^1$. We need to adjust the exponents of both 2 and 3 to be divisible by 3. To do this:

• Increase the exponent of 2 from 2 to 3 (since $3 - 2 = 1$)
• Increase the exponent of 3 from 1 to 3 (since $3 - 1 = 2$)

Thus, the smallest number that is a multiple of 3, 4, and 6, and is also a perfect cube, will be $2^3 \times 3^3 = 8 \times 27 = 216$.

Step 3: Cube Root of 216

The cube root of 216 is: [ \sqrt[3]{216} = \sqrt[3]{2^3 \times 3^3} = 2 \times 3 = 6 ]

Step 4: Check if the cube root is divisible by 3, 4, and 6

The cube root is 6.

• 6 is divisible by 3.
• 6 is not divisible by 4 (since $6 \div 4 = 1.5$).
• 6 is divisible by 6.

Thus, the cube root (6) is divisible by 3 and 6, but not divisible by 4.

Final Answer:

• The smallest multiple of 3, 4, and 6 that is a perfect cube is 216.
• The cube root of 216 is 6.
• The cube root (6) is divisible by 3 and 6, but not by 4.

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Do you want further details or have any questions?

Here are some related questions for further exploration:

1. What is the least common multiple (LCM) of 3, 4, and 6?
2. What is the smallest perfect square that is divisible by 3, 4, and 6?
3. What is the cube root of the least common multiple of 3, 4, and 6?
4. How can you find the greatest common divisor (GCD) of 3, 4, and 6?
5. What are the prime factorizations of other small numbers that are multiples of 3, 4, and 6?

Tip: When dealing with powers and perfect cubes or squares, always check the exponents in prime factorizations to determine the necessary adjustments!