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!