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We are asked to express the sum \( z + \frac{z}{6} + \frac{6}{z} \) as a single fraction and simplify.

### Step-by-step solution:

1. **Find a common denominator:**

   To combine these terms, we first need to find a common denominator. The terms are \( z \), \( \frac{z}{6} \), and \( \frac{6}{z} \). We can treat \( z \) as \( \frac{z}{1} \).

   The least common denominator (LCD) of the fractions is \( 6z \) (since the denominators are \( 1 \), \( 6 \), and \( z \)).

2. **Rewrite each term with the LCD:**
   - \( z = \frac{z \cdot 6z}{1 \cdot 6z} = \frac{6z^2}{6z} \)
   - \( \frac{z}{6} = \frac{z \cdot z}{6 \cdot z} = \frac{z^2}{6z} \)
   - \( \frac{6}{z} = \frac{6 \cdot 6}{z \cdot 6} = \frac{36}{6z} \)

3. **Combine all terms into a single fraction:**

   Now that all the terms have the same denominator, we can add them:
   \[
   \frac{6z^2}{6z} + \frac{z^2}{6z} + \frac{36}{6z} = \frac{6z^2 + z^2 + 36}{6z}
   \]
   Simplifying the numerator:
   \[
   6z^2 + z^2 + 36 = 7z^2 + 36
   \]

4. **Final expression:**
   So, the expression as a single fraction is:
   \[
   \frac{7z^2 + 36}{6z}
   \]

This is the simplified form of the expression.

Let me know if you would like further details or have any questions!

### Related questions:
1. How do you find the least common denominator of three terms?
2. How would this process change if there were a fourth term with a different denominator?
3. Can you factor the numerator \(7z^2 + 36\)?
4. How do you simplify an algebraic expression with mixed terms like \( z \) and constants?
5. What are the steps to solve complex rational expressions?

### Tip:
When simplifying expressions with fractions, always start by finding the least common denominator (LCD) to combine all terms easily.

Write the expression z + z/6 + 6/z as a single fraction and simplify.

Learn how to simplify the algebraic expression z + z/6 + 6/z into a single fraction. This step-by-step solution helps students understand how to find a common denominator and simplify complex rational expressions. Suitable for Grades 8-10.

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