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!

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.