Question

1. multiply \\(-2x^{-3}y(5yx^5 + 8xy - 4y^2x^2)\\).

\\(-10x^2y^2 - 16x^{-2}y^2 + 8x^{-1}y^3\\)

\\(-28x^{-1}y^7\\)

\\(-10x^{-2}y^6 - 16x^{-2}y^2 + 8x^{-1}y^3\\)

\\(-10x^2y^2 - 16x^{-2}y^2 - 8x^{-1}y^3\\)

Explanation:

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<reasoning>

Distribute the monomial to each term in the trinomial

\[
-2x^{-3}y(5yx^5 + 8xy - 4y^2x^2) = (-2x^{-3}y)(5yx^5) + (-2x^{-3}y)(8xy) - (-2x^{-3}y)(4y^2x^2)
\]

Apply exponent rules to simplify each product

\[

$$\begin{aligned} (-2x^{-3}y)(5yx^5) &= -10x^{-3+5}y^{1+1} = -10x^2y^2 \\ (-2x^{-3}y)(8xy) &= -16x^{-3+1}y^{1+1} = -16x^{-2}y^2 \\ -(-2x^{-3}y)(4y^2x^2) &= 8x^{-3+2}y^{1+2} = 8x^{-1}y^3 \end{aligned}$$

\]

Combine the simplified terms

\[
-10x^2y^2 - 16x^{-2}y^2 + 8x^{-1}y^3
\]
</reasoning>

<answer>
<mcq-correct>(A) \(-10x^2y^2 - 16x^{-2}y^2 + 8x^{-1}y^3\)</mcq-correct>
<mcq-option>(B) \(-28x^{-1}y^7\)</mcq-option>
<mcq-option>(C) \(-10x^{-2}y^6 - 16x^{-2}y^2 + 8x^{-1}y^3\)</mcq-option>
<mcq-option>(D) \(-10x^2y^2 - 16x^{-2}y^2 - 8x^{-1}y^3\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Distribute the monomial to each term in the trinomial

\[
-2x^{-3}y(5yx^5 + 8xy - 4y^2x^2) = (-2x^{-3}y)(5yx^5) + (-2x^{-3}y)(8xy) - (-2x^{-3}y)(4y^2x^2)
\]

Apply exponent rules to simplify each product

\[

$$\begin{aligned} (-2x^{-3}y)(5yx^5) &= -10x^{-3+5}y^{1+1} = -10x^2y^2 \\ (-2x^{-3}y)(8xy) &= -16x^{-3+1}y^{1+1} = -16x^{-2}y^2 \\ -(-2x^{-3}y)(4y^2x^2) &= 8x^{-3+2}y^{1+2} = 8x^{-1}y^3 \end{aligned}$$

\]

Combine the simplified terms

\[
-10x^2y^2 - 16x^{-2}y^2 + 8x^{-1}y^3
\]
</reasoning>

<answer>
<mcq-correct>(A) \(-10x^2y^2 - 16x^{-2}y^2 + 8x^{-1}y^3\)</mcq-correct>
<mcq-option>(B) \(-28x^{-1}y^7\)</mcq-option>
<mcq-option>(C) \(-10x^{-2}y^6 - 16x^{-2}y^2 + 8x^{-1}y^3\)</mcq-option>
<mcq-option>(D) \(-10x^2y^2 - 16x^{-2}y^2 - 8x^{-1}y^3\)</mcq-option>
</answer>

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