Simplify the rational expressions below.

1.) 15 

      __

     25

2.) 3x³

      __

    12x²

3.) 6m-3n 

      _______

        2m-n

 4.) x²-9

     ______

       x-3

5.) a²+2a

    ________

      2a+4

Let's simplify each rational expression step by step.

### 1) \(\frac{15}{25}\)
Both 15 and 25 can be simplified by their greatest common divisor (GCD), which is 5.
\[
\frac{15}{25} = \frac{15 \div 5}{25 \div 5} = \frac{3}{5}
\]
**Simplified Expression**: \(\frac{3}{5}\)

---

### 2) \(\frac{3x^3}{12x^2}\)
First, simplify the numerical coefficients \(\frac{3}{12} = \frac{1}{4}\).
Next, simplify the variable part by subtracting exponents: \(x^3 \div x^2 = x^{3-2} = x^1 = x\).
\[
\frac{3x^3}{12x^2} = \frac{1}{4}x
\]
**Simplified Expression**: \(\frac{x}{4}\)

---

### 3) \(\frac{6m - 3n}{2m - n}\)
Factor out the greatest common factor from the numerator:
\[
6m - 3n = 3(2m - n)
\]
Now the expression becomes:
\[
\frac{3(2m - n)}{2m - n}
\]
Since \(2m - n\) is the same in both the numerator and denominator, they cancel out:
\[
\frac{3(2m - n)}{2m - n} = 3
\]
**Simplified Expression**: 3

---

### 4) \(\frac{x^2 - 9}{x - 3}\)
This is a difference of squares. Recall that \(a^2 - b^2 = (a - b)(a + b)\).
\[
x^2 - 9 = (x - 3)(x + 3)
\]
Now the expression becomes:
\[
\frac{(x - 3)(x + 3)}{x - 3}
\]
Cancel the \(x - 3\) terms:
\[
\frac{(x - 3)(x + 3)}{x - 3} = x + 3
\]
**Simplified Expression**: \(x + 3\)

---

### 5) \(\frac{a^2 + 2a}{2a + 4}\)
Factor out the greatest common factors from both the numerator and the denominator:
\[
a^2 + 2a = a(a + 2), \quad 2a + 4 = 2(a + 2)
\]
Now the expression becomes:
\[
\frac{a(a + 2)}{2(a + 2)}
\]
Cancel the \(a + 2\) terms:
\[
\frac{a(a + 2)}{2(a + 2)} = \frac{a}{2}
\]
**Simplified Expression**: \(\frac{a}{2}\)

---

Let me know if you want details on any of the steps! Here are some related questions to explore:

1. How do you factor expressions like the difference of squares?
2. What is the process for simplifying rational expressions involving variables?
3. How do you find the greatest common factor of polynomials?
4. When can terms in rational expressions be canceled?
5. What is the rule for subtracting exponents in division?

**Tip**: Always factor both the numerator and denominator fully before canceling terms to avoid mistakes!

Simplify the rational expressions below.

1.) 15/25
2.) 3x³/12x²
3.) (6m - 3n)/(2m - n)
4.) (x² - 9)/(x - 3)
5.) (a² + 2a)/(2a + 4)

Learn how to simplify rational expressions using methods like GCD, factoring, and the difference of squares. This step-by-step solution covers problems involving both numerical and algebraic fractions, suitable for Grades 9-11.

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