https://file.aiquickdraw.com/mathbot/file/17262991885434y9uc9vd.jpg

Let's solve the problem step by step for both numbers.

### (i) Find the smallest number by which 300 should be multiplied to obtain a perfect cube:

1. **Prime factorization of 300**:
   \[
   300 = 2^2 \times 3^1 \times 5^2
   \]
   
   To make this a perfect cube, the exponents of all prime factors must be divisible by 3. Let's adjust the exponents of each factor:
   - For \(2^2\), we need one more factor of 2 to make \(2^3\).
   - For \(3^1\), we need two more factors of 3 to make \(3^3\).
   - For \(5^2\), we need one more factor of 5 to make \(5^3\).

   Therefore, the smallest number we need to multiply 300 by is:
   \[
   2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90
   \]

### (ii) Find the smallest number by which 1125 should be multiplied to obtain a perfect cube:

1. **Prime factorization of 1125**:
   \[
   1125 = 3^2 \times 5^3
   \]
   
   To make this a perfect cube, the exponents of all prime factors must be divisible by 3:
   - For \(3^2\), we need one more factor of 3 to make \(3^3\).
   - \(5^3\) is already a perfect cube, so no changes are needed for the factor of 5.

   Therefore, the smallest number we need to multiply 1125 by is:
   \[
   3^1 = 3
   \]

### Final Answer:
- For 300, the smallest number to multiply is **90**.
- For 1125, the smallest number to multiply is **3**.

Let me know if you need more details or have any questions!

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Here are some related questions to explore further:
1. What is the prime factorization of another number, like 450?
2. How do we determine if a number is a perfect cube?
3. Can you find the smallest number by which 48 should be multiplied to become a perfect cube?
4. What are the steps for finding the smallest number by which 720 should be multiplied for a perfect cube?
5. How can prime factorization help in solving cube root problems?

**Tip:** Always ensure the exponents in prime factorization are divisible by 3 for perfect cubes.

For each of the following numbers, find the smallest number by which it should be multiplied to obtain a perfect cube. (i) 300 (ii) 1125

Learn how to find the smallest number by which 300 and 1125 should be multiplied to obtain a perfect cube. Explore prime factorization and cube rules for detailed solutions, suitable for Grades 8-10 students.

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