Question

related questions: lesson 2: expressions

q. which of the following statements demonstrates how the distributive property can be used to find the product of \\(4 \cdot 76\\)?

a. \\(76 \cdot 4\\)
b. \\((4 + 70) \cdot (4 + 6)\\)
c. \\(4(7 + 6) = 4 \cdot 7 + 4 \cdot 6\\)
d. \\(4(70 + 6) = 4 \cdot 70 + 4 \cdot 6\\)

Explanation:

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<reasoning>

Analyze the given product

We need to find the product \(4 \cdot 76\). To use the distributive property, we first decompose one of the factors into a sum of its place values. The number \(76\) can be written as \(70 + 6\).

Apply the distributive property

The distributive property states that \(a(b + c) = a \cdot b + a \cdot c\). Substituting \(a = 4\), \(b = 70\), and \(c = 6\), we get:
\[
4(70 + 6) = 4 \cdot 70 + 4 \cdot 6
\]

Evaluate the given options

• Option A shows \(76 \cdot 4\), which demonstrates the commutative property of multiplication.
• Option B shows \((4 + 70) \cdot (4 + 6)\), which does not equal \(4 \cdot 76\).
• Option C shows \(4(7 + 6) = 4 \cdot 7 + 4 \cdot 6\), which represents \(4 \cdot 13\), not \(4 \cdot 76\).
• Option D shows \(4(70 + 6) = 4 \cdot 70 + 4 \cdot 6\), which correctly represents \(4 \cdot 76\) using the distributive property.

</reasoning>

<answer>
<mcq-option>(A) \(76 \cdot 4\)</mcq-option>
<mcq-option>(B) \((4 + 70) \cdot (4 + 6)\)</mcq-option>
<mcq-option>(C) \(4(7 + 6) = 4 \cdot 7 + 4 \cdot 6\)</mcq-option>
<mcq-correct>(D) \(4(70 + 6) = 4 \cdot 70 + 4 \cdot 6\)</mcq-correct>
</answer>

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Answer:

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<reasoning>

Analyze the given product

We need to find the product \(4 \cdot 76\). To use the distributive property, we first decompose one of the factors into a sum of its place values. The number \(76\) can be written as \(70 + 6\).

Apply the distributive property

The distributive property states that \(a(b + c) = a \cdot b + a \cdot c\). Substituting \(a = 4\), \(b = 70\), and \(c = 6\), we get:
\[
4(70 + 6) = 4 \cdot 70 + 4 \cdot 6
\]

Evaluate the given options

• Option A shows \(76 \cdot 4\), which demonstrates the commutative property of multiplication.
• Option B shows \((4 + 70) \cdot (4 + 6)\), which does not equal \(4 \cdot 76\).
• Option C shows \(4(7 + 6) = 4 \cdot 7 + 4 \cdot 6\), which represents \(4 \cdot 13\), not \(4 \cdot 76\).
• Option D shows \(4(70 + 6) = 4 \cdot 70 + 4 \cdot 6\), which correctly represents \(4 \cdot 76\) using the distributive property.

</reasoning>

<answer>
<mcq-option>(A) \(76 \cdot 4\)</mcq-option>
<mcq-option>(B) \((4 + 70) \cdot (4 + 6)\)</mcq-option>
<mcq-option>(C) \(4(7 + 6) = 4 \cdot 7 + 4 \cdot 6\)</mcq-option>
<mcq-correct>(D) \(4(70 + 6) = 4 \cdot 70 + 4 \cdot 6\)</mcq-correct>
</answer>

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