https://file.aiquickdraw.com/mathbot/file/1729307042158h3vk1mwc.jpg

Let's simplify the given rational expressions step by step.

### 3. \(\dfrac{2x - 4y}{x^2 - 4y^2}\)

- The numerator is \(2x - 4y\). We can factor out the common factor of 2:  
  \[ 2(x - 2y) \]
- The denominator is \(x^2 - 4y^2\), which is a difference of squares:  
  \[ x^2 - (2y)^2 = (x - 2y)(x + 2y) \]
  
Thus, the expression becomes:
\[
\dfrac{2(x - 2y)}{(x - 2y)(x + 2y)}
\]
Cancel out the common factor \((x - 2y)\):
\[
\dfrac{2}{x + 2y}
\]

### 4. \(\dfrac{12x - 12y}{18y - 18x}\)

- The numerator is \(12x - 12y\), which can be factored as:
  \[ 12(x - y) \]
- The denominator is \(18y - 18x\), which can be factored as:
  \[ 18(y - x) \]
  Notice that \(y - x\) is the negative of \(x - y\):
  \[ 18(y - x) = -18(x - y) \]

So the expression becomes:
\[
\dfrac{12(x - y)}{-18(x - y)}
\]
Cancel out \((x - y)\), and you're left with:
\[
\dfrac{12}{-18} = -\dfrac{2}{3}
\]

### 5. \(\dfrac{3x + 6}{5x + 10}\)

- The numerator is \(3x + 6\), which can be factored as:
  \[ 3(x + 2) \]
- The denominator is \(5x + 10\), which can be factored as:
  \[ 5(x + 2) \]

Thus, the expression becomes:
\[
\dfrac{3(x + 2)}{5(x + 2)}
\]
Cancel out \((x + 2)\), and you're left with:
\[
\dfrac{3}{5}
\]

---

Do you want further details or have any questions?

Here are 5 related questions:
1. How do you identify common factors in a rational expression?
2. What are the key steps for simplifying a difference of squares?
3. When can we cancel out factors in rational expressions?
4. Why does changing the order of subtraction introduce a negative sign?
5. How do you simplify a rational expression when both terms are linear?

**Tip**: Always check for common factors in both the numerator and denominator before canceling to avoid errors in simplification.

Simplify the given rational expressions by reducing to lowest terms. Assume all denominators are non-zero. Examples: (2x - 4y) / (x^2 - 4y^2), (12x - 12y) / (18y - 18x), (3x + 6) / (5x + 10).

Learn how to simplify rational expressions such as (2x - 4y) / (x^2 - 4y^2) by factoring and reducing to lowest terms. This guide includes step-by-step solutions for expressions involving common factors and differences of squares. Suitable for high school students (Grades 9-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation