We are given a geometric sequence $\{a_n\}$ such that $a_5 = 1$ and $a_8 = 8$. We are asked to find the common ratio $q$ of this geometric sequence.

1. Problem Description

We are given a geometric sequence $\{a_n\}$ such that $a_5 = 1$ and $a_8 = 8$. We are asked to find the common ratio $q$ of this geometric sequence.

2. Solution Steps

In a geometric sequence, the $n$-th term can be written as $a_n = a_1 q^{n-1}$, where $a_1$ is the first term and $q$ is the common ratio.

We are given $a_5 = 1$ and $a_8 = 8$.

So we have:

$a_5 = a_1 q^{5-1} = a_1 q^4 = 1$

$a_8 = a_1 q^{8-1} = a_1 q^7 = 8$

Dividing the second equation by the first equation, we get:

$\frac{a_1 q^7}{a_1 q^4} = \frac{8}{1}$

$q^3 = 8$

Taking the cube root of both sides, we get:

$q = \sqrt[3]{8} = 2$

3. Final Answer

The common ratio $q = 2$.