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To factor the expression \( 54x^3 - 250y^3 \) as a difference of two cubes, let’s proceed with the following steps:

1. **Identify it as a difference of cubes**:
   The difference of two cubes formula is:
   \[
   a^3 - b^3 = (a - b)(a^2 + ab + b^2)
   \]
   We need to express \( 54x^3 \) and \( 250y^3 \) as cubes of certain terms.

2. **Rewrite the terms as cubes**:
   - \( 54x^3 = (3x)^3 \)
   - \( 250y^3 = (5y)^3 \)

   So, the expression \( 54x^3 - 250y^3 \) can be rewritten as:
   \[
   (3x)^3 - (5y)^3
   \]

3. **Apply the difference of cubes formula**:
   Using \( a = 3x \) and \( b = 5y \), we apply the formula:
   \[
   (3x)^3 - (5y)^3 = (3x - 5y)((3x)^2 + (3x)(5y) + (5y)^2)
   \]

4. **Simplify each term in the factored form**:
   - \( (3x)^2 = 9x^2 \)
   - \( (3x)(5y) = 15xy \)
   - \( (5y)^2 = 25y^2 \)

   Substituting these into the factored form, we get:
   \[
   (3x - 5y)(9x^2 + 15xy + 25y^2)
   \]

### Final Answer:
The factored form of \( 54x^3 - 250y^3 \) is:
\[
(3x - 5y)(9x^2 + 15xy + 25y^2)
\]

Would you like more details or have any questions?

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Here are five related questions to expand your understanding:

1. How do you factor a sum of cubes, like \( a^3 + b^3 \)?
2. What is the process for factoring higher powers, such as \( a^5 - b^5 \)?
3. How would the factorization change if this were a sum of cubes instead?
4. How can you verify that the factorization is correct by expanding the factors?
5. What are some practical applications of factoring differences of cubes in real-world problems?

**Tip:** When factoring expressions, always check for common factors first before using special formulas like the difference of cubes.

Factor the following difference of two cubes: 54x^3 - 250y^3

Learn how to factor the expression 54x^3 - 250y^3 using the difference of cubes formula. This algebra problem is suitable for high school students in Grades 9-12 and covers key concepts in factoring and special binomials.

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