Factor the following 25/36x²y⁴-36x⁴y²

To factor the expression \( \frac{25}{36}x^2y^4 - 36x^4y^2 \), follow these steps:

1. **Identify and factor out the greatest common factor (GCF):**
   \[ \text{GCF of } \frac{25}{36}x^2y^4 \text{ and } 36x^4y^2 \text{ is } x^2y^2 \]

   Thus, we can factor \( x^2y^2 \) out of the expression:
   \[ \frac{25}{36}x^2y^4 - 36x^4y^2 = x^2y^2 \left( \frac{25}{36}y^2 - 36x^2 \right) \]

2. **Simplify the expression inside the parentheses:**
   \[ \frac{25}{36}y^2 - 36x^2 \]

3. **Find a common denominator and rewrite the terms:**
   The least common multiple (LCM) of 36 and 1 (to bring \( 36x^2 \) to a common denominator) is 36:
   \[ \frac{25}{36}y^2 - \frac{36 \times 36}{36}x^2 = \frac{25y^2 - 1296x^2}{36} \]

4. **Recognize and apply the difference of squares formula:**
   \[ a^2 - b^2 = (a - b)(a + b) \]

   Here, \( a = 5y \) and \( b = 36x \):
   \[ 25y^2 - 1296x^2 = (5y)^2 - (36x)^2 = (5y - 36x)(5y + 36x) \]

5. **Combine the factored terms:**
   \[ x^2y^2 \left( \frac{(5y - 36x)(5y + 36x)}{36} \right) \]

6. **Simplify the expression:**
   \[ \boxed{x^2y^2 \cdot \frac{(5y - 36x)(5y + 36x)}{36}} \]

Thus, the factored form of \( \frac{25}{36}x^2y^4 - 36x^4y^2 \) is:
\[ \boxed{\frac{x^2y^2 (5y - 36x)(5y + 36x)}{36}} \]

Would you like more details or have any questions about this solution?

Here are some related questions you might find helpful:
1. How do you identify the greatest common factor in an algebraic expression?
2. What are the steps to factor a difference of squares?
3. How do you simplify an algebraic expression with fractions?
4. What is the process to find a common denominator in algebraic fractions?
5. How can recognizing common algebraic identities help in factoring complex expressions?
6. What are some tips for checking if your factored expression is correct?
7. How can the distributive property be used in reverse to factor expressions?
8. What are other examples of factoring algebraic expressions with multiple variables?

**Tip:** When factoring expressions, always look for a common factor first; it can simplify the process significantly.

Learn how to factor the expression 25/36x²y⁴ - 36x⁴y² using step-by-step instructions. This problem involves factoring algebraic expressions with multiple variables, including the application of the difference of squares formula. Suitable for high school students.

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