The given sequence is:

$-9x^8, \quad 36x^{12}, \quad -144x^{16}, \quad \dots$

This is a geometric sequence where each term is obtained by multiplying the previous term by a constant ratio $r$. Let's first find the common ratio $r$ by dividing the second term by the first term:

$r = \frac{36x^{12}}{-9x^8} = \frac{36}{-9} \cdot \frac{x^{12}}{x^8} = -4x^4$

Thus, the common ratio $r = -4x^4$.

To find the 12th term of the sequence, we use the formula for the $n$-th term of a geometric sequence:

$a_n = a_1 \cdot r^{n-1}$

Here:

• $a_1 = -9x^8$
• $r = -4x^4$
• $n = 12$

Now, we calculate the 12th term:

$a_{12} = -9x^8 \cdot (-4x^4)^{11}$

Let's break this calculation down:

1. Calculate the power of $r$: $(-4x^4)^{11} = (-4)^{11} \cdot (x^4)^{11} = -4194304 \cdot x^{44}$

2. Now multiply by $a_1$: $a_{12} = -9x^8 \cdot (-4194304x^{44}) = 9 \cdot 4194304 \cdot x^{8+44} = 37748736x^{52}$

Thus, the 12th term of the sequence is:

$a_{12} = 37748736x^{52}$

Would you like further details or have any questions about this solution?

Here are five related questions to explore:

1. What is the common ratio of a geometric sequence?
2. How do you find the general term in a geometric sequence?
3. What happens to the terms of a geometric sequence if the common ratio is negative?
4. How can geometric sequences be applied in real-life problems?
5. What is the sum of the first $n$ terms in a geometric sequence?

Tip: In a geometric sequence, the common ratio can be found by dividing any term by its preceding term.