https://file.aiquickdraw.com/mathbot/file/1729524013492o3j9pvjs.jpg

The expression in the image is:

\[
\left( \frac{z+6}{3z+9} - \frac{1}{z+3} \right) \div \frac{z+2}{27z}
\]

Let’s simplify this step by step.

### Step 1: Simplify the first part within parentheses
The first part of the expression inside the parentheses is:

\[
\frac{z+6}{3z+9} - \frac{1}{z+3}
\]

Notice that \( 3z+9 \) can be factored as \( 3(z+3) \). So, the first term becomes:

\[
\frac{z+6}{3(z+3)}
\]

Now, the expression becomes:

\[
\frac{z+6}{3(z+3)} - \frac{1}{z+3}
\]

The least common denominator (LCD) is \( 3(z+3) \). Rewrite the second term with this denominator:

\[
\frac{1}{z+3} = \frac{3}{3(z+3)}
\]

Now, subtract the fractions:

\[
\frac{z+6}{3(z+3)} - \frac{3}{3(z+3)} = \frac{(z+6) - 3}{3(z+3)} = \frac{z+3}{3(z+3)}
\]

Notice that \( z+3 \) cancels out in the numerator and denominator:

\[
\frac{z+3}{3(z+3)} = \frac{1}{3}
\]

So, the expression inside the parentheses simplifies to \( \frac{1}{3} \).

### Step 2: Simplify the division
Now the expression is:

\[
\frac{1}{3} \div \frac{z+2}{27z}
\]

Dividing by a fraction is equivalent to multiplying by its reciprocal. So, rewrite the division as a multiplication:

\[
\frac{1}{3} \times \frac{27z}{z+2}
\]

Now, simplify:

\[
\frac{1 \times 27z}{3 \times (z+2)} = \frac{27z}{3(z+2)}
\]

Finally, simplify \( \frac{27z}{3} \):

\[
\frac{27z}{3(z+2)} = \frac{9z}{z+2}
\]

### Final Answer:
The simplified expression is:

\[
\frac{9z}{z+2}
\]

Would you like further details or have any questions on this process?

Here are some related questions:
1. How do you handle division of complex algebraic fractions?
2. What happens when terms cancel out in an algebraic expression?
3. Can this expression be simplified further if \( z = -2 \)?
4. How would the process change if there were multiple variables in the denominator?
5. What does factoring do in simplifying algebraic expressions?

**Tip:** Always look for factoring opportunities in both the numerator and the denominator to simplify complex algebraic fractions!

Simplify the expression: ((z + 6) / (3z + 9) - 1 / (z + 3)) ÷ (z + 2) / 27z

Learn how to simplify the algebraic expression ((z + 6) / (3z + 9) - 1 / (z + 3)) ÷ (z + 2) / 27z with a step-by-step approach. This solution involves factoring, finding a common denominator, and simplifying fractions. Suitable for students in Grades 8-10.

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