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.

Explore the concept of exponents and perfect cubes with an explanation on why the answer to making 27783 a perfect cube is 3² instead of 3³. This problem covers prime factorization, adjusting exponents for perfect powers, and mathematical reasoning suitable for Grades 6-8.

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