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why can I not simplify 2c with c to give c

The expression you've provided is:

\[
\frac{a + 6}{c} \times \frac{2c}{2a + 6}
\]

Let's break down why you cannot simply cancel out the \( c \) and \( 2c \) directly.

### Expression Analysis:
1. **Numerators and Denominators**: The expression is a multiplication of two fractions. The first fraction is \(\frac{a + 6}{c}\) and the second is \(\frac{2c}{2a + 6}\).

2. **Simplifying Terms**: 
   - The \( c \) in the denominator of the first fraction and the \( 2c \) in the numerator of the second fraction **cannot** be canceled directly because the \( 2c \) is part of a larger term within a fraction. Specifically, the \( 2c \) is part of the product \( 2c \), and the denominator \( c \) is not just a standalone term but part of the entire fraction.

3. **Cancellation Rules**: 
   - To cancel terms across fractions, those terms must be either factored out entirely from both the numerator and denominator or multiplied separately. In this case, \( 2c \) and \( c \) are in different terms across the multiplication and are not common factors across the entire expression.

### Proper Simplification:

However, what you can do is simplify each fraction independently:

- Start with the second fraction:
  \[
  \frac{2c}{2a + 6} 
  \]
  Factor out the common factor of 2 from the denominator:
  \[
  = \frac{2c}{2(a + 3)} = \frac{c}{a + 3}
  \]

So, the original expression can be rewritten as:
\[
\frac{a + 6}{c} \times \frac{c}{a + 3}
\]

Now, you can **cancel** the \( c \) in the numerator of the second fraction with the \( c \) in the denominator of the first fraction:

\[
= \frac{a + 6}{a + 3}
\]

### Final Answer:
After simplification, the expression reduces to:

\[
\frac{a + 6}{a + 3}
\]

This is the fully simplified form, and it shows that the cancellation should happen after appropriate factoring, not before.

---

Would you like further details or have any additional questions? 

Here are some related questions that might interest you:
1. How do you factor complex polynomials?
2. What is the general rule for canceling terms in algebraic fractions?
3. Can you always cancel out terms in a fraction multiplication?
4. How do you handle expressions with multiple variables?
5. What are common mistakes in simplifying algebraic expressions?
6. How does the distributive property help in simplifying fractions?
7. Can you simplify expressions with addition or subtraction in the numerator and denominator?
8. How does cross-multiplication work in simplifying fractions?

**Tip:** Always look for common factors across the entire numerator and denominator before canceling any terms. This helps avoid mistakes in simplification.

Explore why you cannot directly simplify the expression 2c with c to give c. Learn about the rules of fraction simplification and why terms must be factored correctly before cancellation. Suitable for advanced high school students and those preparing for algebraic fraction simplification.

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