To determine the value of the angle $z$ formed between the extensions of a regular hexagon and a regular octagon, we can follow these steps:

Step-by-Step Solution

1. Calculate the Interior Angles:

   • A regular hexagon has $6$ sides, and the formula for the interior angle of a regular polygon with $n$ sides is: $\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}$

   • For the hexagon ($n = 6$): $\text{Hexagon interior angle} = \frac{(6-2) \times 180^\circ}{6} = \frac{4 \times 180^\circ}{6} = 120^\circ$

   • For the octagon ($n = 8$): $\text{Octagon interior angle} = \frac{(8-2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = 135^\circ$

2. Determine the Exterior Angles:

   • The exterior angle of a polygon is the supplement of the interior angle, which is calculated by: $\text{Exterior angle} = 180^\circ - \text{Interior angle}$

   • For the hexagon: $\text{Hexagon exterior angle} = 180^\circ - 120^\circ = 60^\circ$

   • For the octagon: $\text{Octagon exterior angle} = 180^\circ - 135^\circ = 45^\circ$

3. Calculate Angle $z$:

   • The angle $z$ is formed by the extension of a side of the hexagon and a side of the octagon.
   • Since $z$ is the angle between these two extensions, it is equal to the sum of the two exterior angles: $z = 60^\circ + 45^\circ = 105^\circ$

Conclusion

The value of $z$ is: $\boxed{105^\circ}$

Would you like more details or have any questions?

Here are some related questions for further practice:

1. What is the sum of all interior angles of a regular octagon?
2. If a regular polygon has an exterior angle of $40^\circ$, how many sides does it have?
3. How would the angle $z$ change if the polygons were a pentagon and a decagon?
4. Explain why the exterior angle formula works for any regular polygon.
5. How can we determine the number of sides in a regular polygon if we only know one interior angle?

Tip: For regular polygons, the interior and exterior angles can always be calculated with the same formulas, regardless of the number of sides.