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write the simplified rational expression version of this

The given expression is:

\[
\frac{2a - b}{4a^2 - b^2}
\]

To simplify this rational expression, we can proceed as follows:

### Step 1: Factor the denominator
The denominator \(4a^2 - b^2\) is a difference of squares, which can be factored using the formula:

\[
x^2 - y^2 = (x - y)(x + y)
\]

So, we factor \(4a^2 - b^2\) as:

\[
4a^2 - b^2 = (2a - b)(2a + b)
\]

### Step 2: Simplify the expression
Now, substitute the factored form of the denominator into the original expression:

\[
\frac{2a - b}{(2a - b)(2a + b)}
\]

Here, we can cancel the common factor \(2a - b\) from the numerator and the denominator:

\[
\frac{\cancel{2a - b}}{\cancel{2a - b}(2a + b)} = \frac{1}{2a + b}
\]

### Final Simplified Expression:
\[
\frac{1}{2a + b}
\]

This is the simplified form of the given rational expression.

Would you like more details on any specific step or have any questions?

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**Related Questions:**

1. How do you factor expressions like \(4a^2 - b^2\)?
2. What is the difference of squares and how is it used in algebra?
3. How do you simplify rational expressions?
4. What are the conditions under which you can cancel terms in a rational expression?
5. How would the expression change if the denominator was \(4a^2 + b^2\) instead?

**Tip:** Always check for common factors between the numerator and the denominator before attempting to simplify a rational expression.

Learn how to simplify the rational expression (2a - b) / (4a^2 - b^2) step-by-step. This problem covers key concepts in algebra, including factoring and cancellation of terms. Suitable for high school students.

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