To explain why $3^2$ is the correct answer, let's revisit the goal: we want to make 27783 a perfect cube by multiplying it by the smallest possible number.

Prime Factorization of 27783

As we previously determined: $27783 = 3^4 \times 7^3$

Requirements for a Perfect Cube

For a number to be a perfect cube, all the exponents in its prime factorization must be multiples of 3. Currently, the exponents are:

• $3^4$: Exponent is 4
• $7^3$: Exponent is 3

Adjusting the Exponents

• For $3^4$:
  • The exponent 4 needs to be increased to the nearest multiple of 3, which is 6. Therefore, we need $3^{6-4} = 3^2$.
• For $7^3$:
  • The exponent 3 is already a multiple of 3, so no adjustment is needed.

Multiplication Factor

To make the exponent of 3 a multiple of 3, we need: $3^2$

So, the smallest number by which 27783 should be multiplied to make it a perfect cube is $3^2 = 9$.

Therefore, the smallest number is $\boxed{9}$.

Would you like more details or have any questions?

Here are 8 related questions:

1. How do we determine the prime factorization of a number?
2. Why must the exponents in the prime factorization be multiples of 3 for a perfect cube?
3. What is the difference between perfect squares, perfect cubes, and other perfect powers?
4. How can you verify that a number is a perfect cube?
5. What are some examples of perfect cubes?
6. How does the process change for finding the smallest number to multiply to get a perfect fourth power?
7. Can you explain the significance of the smallest multiplier in other mathematical contexts?
8. How can prime factorization be used in other areas of mathematics?

Tip: Always ensure that all exponents in the prime factorization of a perfect power are adjusted to the required multiple, not just the smallest necessary increment.