By what number should 4608 be divided to get a perfect cube? 

To find the number by which 4608 should be divided to get a perfect cube, we must factorize 4608 and then adjust the factors so that each prime factor appears in multiples of 3, since a perfect cube has every prime factor raised to an exponent that is a multiple of 3.

### Step 1: Prime factorization of 4608

Let's factorize 4608:

\[
4608 \div 2 = 2304
\]
\[
2304 \div 2 = 1152
\]
\[
1152 \div 2 = 576
\]
\[
576 \div 2 = 288
\]
\[
288 \div 2 = 144
\]
\[
144 \div 2 = 72
\]
\[
72 \div 2 = 36
\]
\[
36 \div 2 = 18
\]
\[
18 \div 2 = 9
\]
\[
9 \div 3 = 3
\]
\[
3 \div 3 = 1
\]

So, the prime factorization of 4608 is:

\[
4608 = 2^8 \times 3^2
\]

### Step 2: Adjusting for a perfect cube

To make 4608 a perfect cube, the exponents of the prime factors must be multiples of 3.

- The exponent of 2 is 8, which is not a multiple of 3. To make it a multiple of 3, we need to divide 4608 by \( 2^2 = 4 \) (since \( 8 - 6 = 2 \)).
- The exponent of 3 is 2, which is not a multiple of 3. To make it a multiple of 3, we need to divide 4608 by \( 3^2 = 9 \).

### Step 3: Final division

Now, we need to divide 4608 by the product of \( 2^2 \) and \( 3^2 \):

\[
2^2 \times 3^2 = 4 \times 9 = 36
\]

Thus, 4608 should be divided by **36** to get a perfect cube.

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### Related Questions:
1. What is a perfect cube and how is it defined?
2. Can you explain prime factorization in more detail?
3. How do you verify if a number is a perfect cube?
4. What are other examples of numbers that need adjustments to become perfect cubes?
5. How would you factorize larger numbers efficiently?

#### Tip: 
A perfect cube is a number that can be expressed as \( n^3 \), where \( n \) is an integer. Each prime factor's exponent must be divisible by 3.

By what number should 4608 be divided to get a perfect cube?

Learn how to find the number by which 4608 must be divided to become a perfect cube. This detailed guide uses prime factorization and the perfect cube rule to explain the solution, suitable for students in Grades 8-10.

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