Express as a single fraction in its simplest form x^4-y^4/x^2-y^2

To express the given expression \(\frac{x^4 - y^4}{x^2 - y^2}\) as a single fraction in its simplest form, let's follow these steps:

### Step 1: Factorize the numerator and the denominator.

1. **Numerator:** \(x^4 - y^4\) is a difference of squares:
   \[
   x^4 - y^4 = (x^2)^2 - (y^2)^2 = (x^2 - y^2)(x^2 + y^2).
   \]

2. **Denominator:** \(x^2 - y^2\) is also a difference of squares:
   \[
   x^2 - y^2 = (x - y)(x + y).
   \]

### Step 2: Substitute the factored forms back into the expression.

Substituting the factored forms, we get:
\[
\frac{(x^2 - y^2)(x^2 + y^2)}{(x - y)(x + y)}.
\]

### Step 3: Simplify by canceling common factors.

Since \(x^2 - y^2 = (x - y)(x + y)\), we can cancel these terms from the numerator and denominator:
\[
\frac{\cancel{(x - y)(x + y)}(x^2 + y^2)}{\cancel{(x - y)(x + y)}} = x^2 + y^2.
\]

### Final Answer:
\[
\frac{x^4 - y^4}{x^2 - y^2} = x^2 + y^2.
\]

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

### Related Questions:
1. How do you simplify \(\frac{x^6 - y^6}{x^3 - y^3}\)?
2. What is the simplest form of \(\frac{x^4 - 1}{x^2 - 1}\)?
3. How do you factorize the difference of cubes, like \(x^3 - y^3\)?
4. Can you express \(\frac{a^4 - b^4}{a - b}\) as a single fraction?
5. How do you factor higher powers like \(x^8 - y^8\)?

**Tip:** Remember that recognizing patterns like the difference of squares or cubes can greatly simplify algebraic expressions.

Express as a single fraction in its simplest form \(\frac{x^4 - y^4}{x^2 - y^2}\).

Learn how to simplify the algebraic fraction \(\frac{x^4 - y^4}{x^2 - y^2}\) by factoring polynomials using the difference of squares method. This detailed step-by-step solution explains key concepts in algebra and is suitable for students in Grades 9-12.

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