Question

which of the following represents \\(\sqrt9{x^7}\\) in exponential form?

\\(x^{\frac{9}{7}}\\)

\\(x^{\frac{7}{9}}\\)

\\(9x^7\\)

\\(7x^9\\)

question 8 (multiple choice worth 6 points)
(01.08r hc)

find the volume of a rectangular prism if the length is \\(4x\\), the width is \\(2x\\), and the height is \\(x^3 + 3x + 6\\). use the formula \\(v = l \cdot w \cdot h\\), where \\(l\\) is length, \\(w\\) is width, and \\(h\\) is height, to find the volume.

\\(6x^5 + 18x^3 + 36x^2\\)

\\(6x^6 + 18x^3 + 36x^2\\)

\\(8x^5 + 24x^3 + 48x^2\\)

\\(8x^6 + 24x^3 + 48x^2\\)

Explanation:

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<reasoning>

Convert radical to exponential form

\[
\sqrt[9]{x^7} = x^{\frac{7}{9}}
\]

Multiply length and width

\[
l \cdot w = (4x) \cdot (2x) = 8x^2
\]

Multiply by height to find volume

\[
V = (8x^2) \cdot (x^3 + 3x + 6) = 8x^5 + 24x^3 + 48x^2
\]
</reasoning>

<answer>

Question 1

<mcq-option>(A) \(x^{\frac{9}{7}}\)</mcq-option>
<mcq-correct>(B) \(x^{\frac{7}{9}}\)</mcq-correct>
<mcq-option>(C) \(9x^7\)</mcq-option>
<mcq-option>(D) \(7x^9\)</mcq-option>

Question 2

<mcq-option>(A) \(6x^5 + 18x^3 + 36x^2\)</mcq-option>
<mcq-option>(B) \(6x^6 + 18x^3 + 36x^2\)</mcq-option>
<mcq-correct>(C) \(8x^5 + 24x^3 + 48x^2\)</mcq-correct>
<mcq-option>(D) \(8x^6 + 24x^3 + 48x^2\)</mcq-option>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Rational Exponents and Polynomial Multiplication"
]
}
</post_analysis>

Answer:

<pre_analysis>
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"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
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"concepts_used": [
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"new_concepts": [],
"current_concepts": [
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}
</pre_analysis>

<reasoning>

Convert radical to exponential form

\[
\sqrt[9]{x^7} = x^{\frac{7}{9}}
\]

Multiply length and width

\[
l \cdot w = (4x) \cdot (2x) = 8x^2
\]

Multiply by height to find volume

\[
V = (8x^2) \cdot (x^3 + 3x + 6) = 8x^5 + 24x^3 + 48x^2
\]
</reasoning>

<answer>

Question 1

<mcq-option>(A) \(x^{\frac{9}{7}}\)</mcq-option>
<mcq-correct>(B) \(x^{\frac{7}{9}}\)</mcq-correct>
<mcq-option>(C) \(9x^7\)</mcq-option>
<mcq-option>(D) \(7x^9\)</mcq-option>

Question 2

<mcq-option>(A) \(6x^5 + 18x^3 + 36x^2\)</mcq-option>
<mcq-option>(B) \(6x^6 + 18x^3 + 36x^2\)</mcq-option>
<mcq-correct>(C) \(8x^5 + 24x^3 + 48x^2\)</mcq-correct>
<mcq-option>(D) \(8x^6 + 24x^3 + 48x^2\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Rational Exponents and Polynomial Multiplication"
]
}
</post_analysis>