Let us analyze the problem and solve it step by step.

The goal is to select all expressions equivalent to $-12x^3y^8$.

Given Options

Let’s rewrite the options in standard notation:

• (A): $(2xy)(-6x^2y^5)(y^2)$
• (B): $(-2xy)^2(3xy^6)$
• (C): $-2^2x^3y^3$
• (D): $(-2xy)^2(3xy^6)(y^5)$
• (E): $(-2xy)^2(3xy^2)^2$

We will simplify each expression to see which matches $-12x^3y^8$.

Option (A)

$(2xy)(-6x^2y^5)(y^2)$

• Simplify each factor:
  • $2xy = 2x^1y^1$
  • $-6x^2y^5 = -6x^2y^5$
  • $y^2 = y^2$
• Combine all terms: $(2)(-6)x^{1+2}y^{1+5+2} = -12x^3y^8$ This matches the target expression.

(A) is equivalent.

Option (B)

$(-2xy)^2(3xy^6)$

• Simplify $(-2xy)^2$: $(-2)^2(xy)^2 = 4x^2y^2$
• Multiply with $3xy^6$: $(4x^2y^2)(3xy^6) = (4 \cdot 3)x^{2+1}y^{2+6} = 12x^3y^8$ This does not match $-12x^3y^8$ because the sign is positive.

(B) is not equivalent.

Option (C)

$-2^2x^3y^3$

• Simplify: $-2^2 = -4, \quad x^3 = x^3, \quad y^3 = y^3$
• Combine: $-4x^3y^3$ This does not match $-12x^3y^8$.

(C) is not equivalent.

Option (D)

$(-2xy)^2(3xy^6)(y^5)$

• Simplify $(-2xy)^2$: $(-2)^2(xy)^2 = 4x^2y^2$
• Multiply with $3xy^6$ and $y^5$: $(4x^2y^2)(3xy^6)(y^5) = (4 \cdot 3)x^{2+1}y^{2+6+5} = 12x^3y^{13}$ This does not match $-12x^3y^8$.

(D) is not equivalent.

Option (E)

$(-2xy)^2(3xy^2)^2$

• Simplify $(-2xy)^2$: $(-2)^2(xy)^2 = 4x^2y^2$
• Simplify $(3xy^2)^2$: $3^2(xy^2)^2 = 9x^2y^4$
• Multiply: $(4x^2y^2)(9x^2y^4) = (4 \cdot 9)x^{2+2}y^{2+4} = 36x^4y^6$ This does not match $-12x^3y^8$.

(E) is not equivalent.

Final Answer

The only option equivalent to $-12x^3y^8$ is: $\boxed{A}$

Let me know if you have any questions or want detailed explanations!

Related Questions

1. What are the general rules for simplifying powers of a product like $(ab)^n$?
2. How can you determine the sign of a product when multiple terms are involved?
3. How do you simplify an expression with multiple factors of $x^a$ and $y^b$?
4. What is the difference between squaring a term and distributing exponents across products?
5. How do you verify equivalence in algebraic expressions systematically?

Tip

Always carefully apply exponent rules, especially when negative signs or multiple terms are involved!