Let's break down the expression step-by-step.

The given expression is:

$-3(2x^4)(4x^5y)^2$

Step 1: Simplify $(4x^5y)^2$

When squaring a product, we square each factor individually:

$(4x^5y)^2 = (4)^2 (x^5)^2 (y)^2 = 16x^{10}y^2$

Step 2: Substitute back into the original expression

Now substitute $16x^{10}y^2$ into the original expression:

$-3(2x^4)(16x^{10}y^2)$

Step 3: Multiply the constants

Multiply the constants: $-3 \times 2 \times 16$:

$-3 \times 2 = -6, \quad -6 \times 16 = -96$

Step 4: Multiply the powers of $x$

Use the rule for multiplying powers with the same base: $x^a \times x^b = x^{a+b}$.

$x^4 \times x^{10} = x^{4+10} = x^{14}$

Step 5: Combine the expression

Now combine everything:

$-96x^{14}y^2$

So the simplified expression is:

$-96x^{14}y^2$

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Would you like more details on any of the steps? Or do you have any questions?

Here are some related questions for practice:

1. What happens when you multiply terms with different exponents?
2. How do you handle exponents when raising products to a power?
3. What are some common properties of exponents?
4. How would you simplify the expression $(3x^2y^3)^3$?
5. How can you factor an expression like $-96x^{14}y^2$?

Tip: Always remember to apply the power of a product rule and the product of powers rule when simplifying expressions involving exponents.