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.