The given expression is:

$\left(3a^4b^{-2}\right)^3 \left(a^8b^{-1}\right)$

Let's simplify this step by step:

Step 1: Expand the powers

We need to distribute the exponent $3$ across all terms inside the first set of parentheses:

$\left(3a^4b^{-2}\right)^3 = 3^3 \cdot a^{4 \cdot 3} \cdot b^{-2 \cdot 3} = 27a^{12}b^{-6}$

So, the expression becomes:

$27a^{12}b^{-6} \cdot a^8b^{-1}$

Step 2: Combine like terms

Now, combine the powers of $a$ and $b$:

For $a$: $a^{12} \cdot a^8 = a^{12 + 8} = a^{20}$

For $b$: $b^{-6} \cdot b^{-1} = b^{-6 + (-1)} = b^{-7}$

Step 3: Eliminate negative exponents

To eliminate the negative exponent, rewrite $b^{-7}$ as $\frac{1}{b^7}$:

$27a^{20}b^{-7} = \frac{27a^{20}}{b^7}$

Final Answer:

$\frac{27a^{20}}{b^7}$

Let me know if you'd like further details or have any other questions!

Here are 5 related questions for practice:

1. How would you simplify $(2x^5y^{-3})^4(x^2y^6)$?
2. What happens if the exponent outside the parentheses is negative?
3. Can you apply this process to an expression with more variables, such as $(xy^{-2}z^3)^2(x^2z^{-1})$?
4. How do you handle fractional exponents in expressions like this?
5. How do negative exponents affect the overall fraction?

Tip: Always combine like terms by adding their exponents, and remember to convert negative exponents to positive by moving them to the denominator or numerator.