Question

simplify:
\\((2x^3)^3\\)

• \\(2x^6\\)
• \\(2x^9\\)
• \\(8x^6\\)
• \\(8x^9\\)

question 4

simplify:
\\((3x^5y^{-7})^2\\)

• \\(\frac{9x^{10}}{y^{14}}\\)
• \\(\frac{6x^{10}}{y^{14}}\\)
• \\(\frac{1}{9x^{10}y^{14}}\\)
• \\(9\frac{x^7}{y^9}\\)

Explanation:

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"Negative Exponents"
],
"new_concepts": [],
"current_concepts": [
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</pre_analysis>

<reasoning>

Distribute the outer exponent to each factor in the first expression

\[
(2x^3)^3 = 2^3 \cdot (x^3)^3
\]

Simplify the numerical base and apply the power of a power rule

\[
2^3 = 8
\]
\[
(x^3)^3 = x^{3 \cdot 3} = x^9
\]
\[
(2x^3)^3 = 8x^9
\]

Distribute the outer exponent and apply negative exponent rules to the second expression

\[
(3x^5y^{-7})^2 = 3^2 \cdot (x^5)^2 \cdot (y^{-7})^2
\]
\[
3^2 = 9
\]
\[
(x^5)^2 = x^{5 \cdot 2} = x^{10}
\]
\[
(y^{-7})^2 = y^{-7 \cdot 2} = y^{-14} = \frac{1}{y^{14}}
\]
\[
(3x^5y^{-7})^2 = \frac{9x^{10}}{y^{14}}
\]
</reasoning>

<answer>

Question 1

<mcq-option>(A) \(2x^6\)</mcq-option>
<mcq-option>(B) \(2x^9\)</mcq-option>
<mcq-option>(C) \(8x^6\)</mcq-option>
<mcq-correct>(D) \(8x^9\)</mcq-correct>

Question 2

<mcq-correct>(A) \(\frac{9x^{10}}{y^{14}}\)</mcq-correct>
<mcq-option>(B) \(\frac{6x^{10}}{y^{14}}\)</mcq-option>
<mcq-option>(C) \(\frac{1}{9x^{10}y^{14}}\)</mcq-option>
<mcq-option>(D) \(9\frac{x^7}{y^9}\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Exponent Rules"
]
}
</post_analysis>

Answer:

<pre_analysis>
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"question_count": 2,
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"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
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"concepts_used": [
"Exponent Rules",
"Negative Exponents"
],
"new_concepts": [],
"current_concepts": [
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</pre_analysis>

<reasoning>

Distribute the outer exponent to each factor in the first expression

\[
(2x^3)^3 = 2^3 \cdot (x^3)^3
\]

Simplify the numerical base and apply the power of a power rule

\[
2^3 = 8
\]
\[
(x^3)^3 = x^{3 \cdot 3} = x^9
\]
\[
(2x^3)^3 = 8x^9
\]

Distribute the outer exponent and apply negative exponent rules to the second expression

\[
(3x^5y^{-7})^2 = 3^2 \cdot (x^5)^2 \cdot (y^{-7})^2
\]
\[
3^2 = 9
\]
\[
(x^5)^2 = x^{5 \cdot 2} = x^{10}
\]
\[
(y^{-7})^2 = y^{-7 \cdot 2} = y^{-14} = \frac{1}{y^{14}}
\]
\[
(3x^5y^{-7})^2 = \frac{9x^{10}}{y^{14}}
\]
</reasoning>

<answer>

Question 1

<mcq-option>(A) \(2x^6\)</mcq-option>
<mcq-option>(B) \(2x^9\)</mcq-option>
<mcq-option>(C) \(8x^6\)</mcq-option>
<mcq-correct>(D) \(8x^9\)</mcq-correct>

Question 2

<mcq-correct>(A) \(\frac{9x^{10}}{y^{14}}\)</mcq-correct>
<mcq-option>(B) \(\frac{6x^{10}}{y^{14}}\)</mcq-option>
<mcq-option>(C) \(\frac{1}{9x^{10}y^{14}}\)</mcq-option>
<mcq-option>(D) \(9\frac{x^7}{y^9}\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Exponent Rules"
]
}
</post_analysis>