To simplify the expression $(4ab^2)^3 \cdot (-2a^5)$, let’s break it down step-by-step:

Step 1: Simplify $(4ab^2)^3$

We apply the power to each component inside the parentheses: $(4ab^2)^3 = 4^3 \cdot a^3 \cdot (b^2)^3$ Calculating each part:

• $4^3 = 64$
• $a^3$ remains as $a^3$
• $(b^2)^3 = b^{2 \cdot 3} = b^6$

So, $(4ab^2)^3 = 64a^3b^6$

Step 2: Multiply by $-2a^5$

Now, we take $64a^3b^6$ and multiply it by $-2a^5$: $64a^3b^6 \cdot (-2a^5) = 64 \cdot (-2) \cdot a^{3+5} \cdot b^6$ Calculating each part:

• $64 \cdot (-2) = -128$
• $a^{3+5} = a^8$
• $b^6$ remains as $b^6$

Thus, the expression simplifies to: $-128a^8b^6$

Answer

The correct answer is: $\text{-128a}^8\text{b}^6$

Would you like a detailed explanation of any part of this process, or do you have any questions?

Here are some related questions for practice:

1. How do you simplify $(3x^2y^3)^4 \cdot (-5x^3y)$?
2. What is the result of raising a product to a power, like $(7m^2n^3)^2$?
3. How do you apply the power rule in expressions like $(2a^4b^3)^5$?
4. What happens when multiplying terms with different powers, such as $x^3 \cdot x^7$?
5. How do you simplify expressions with multiple variables raised to powers?

Tip: When raising a term in parentheses to a power, apply the exponent to each factor inside the parentheses separately before performing additional operations.