Question

simplify.

\\(\sqrt{75}\\)

\\(3\sqrt{5}\\)

\\(15\sqrt{5}\\)

\\(25\sqrt{3}\\)

\\(5\sqrt{3}\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Simplifying Radicals",
"Square Roots"
],
"current_concepts": [
"Simplifying Radicals",
"Square Roots"
]
}
</pre_analysis>

<reasoning>

Factor the radicand using the largest perfect square

\[
75 = 25 \times 3
\]

Apply the product property of square roots

\[
\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3}
\]

Simplify the perfect square

\[
\sqrt{25} \times \sqrt{3} = 5\sqrt{3}
\]
</reasoning>

<answer>
<mcq-option>(A) \(3\sqrt{5}\)</mcq-option>
<mcq-option>(B) \(15\sqrt{5}\)</mcq-option>
<mcq-option>(C) \(25\sqrt{3}\)</mcq-option>
<mcq-correct>(D) \(5\sqrt{3}\)</mcq-correct>
</answer>

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

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Simplifying Radicals",
"Square Roots"
],
"current_concepts": [
"Simplifying Radicals",
"Square Roots"
]
}
</pre_analysis>

<reasoning>

Factor the radicand using the largest perfect square

\[
75 = 25 \times 3
\]

Apply the product property of square roots

\[
\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3}
\]

Simplify the perfect square

\[
\sqrt{25} \times \sqrt{3} = 5\sqrt{3}
\]
</reasoning>

<answer>
<mcq-option>(A) \(3\sqrt{5}\)</mcq-option>
<mcq-option>(B) \(15\sqrt{5}\)</mcq-option>
<mcq-option>(C) \(25\sqrt{3}\)</mcq-option>
<mcq-correct>(D) \(5\sqrt{3}\)</mcq-correct>
</answer>

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