Question

multiplying \\(\frac{3}{\sqrt{17}-\sqrt{2}}\\) by which fraction will produce an equivalent fraction with a rational denominator?

\\(\circ\\) \\(\frac{\sqrt{17}-\sqrt{2}}{\sqrt{17}-\sqrt{2}}\\)

\\(\circ\\) \\(\frac{\sqrt{17}+\sqrt{2}}{\sqrt{17}+\sqrt{2}}\\)

\\(\circ\\) \\(\frac{\sqrt{2}-\sqrt{17}}{\sqrt{2}-\sqrt{17}}\\)

\\(\circ\\) \\(\frac{\sqrt{15}}{\sqrt{15}}\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Rationalizing the Denominator",
"Radical Conjugates"
],
"new_concepts": [],
"current_concepts": [
"Rationalizing the Denominator",
"Radical Conjugates"
]
}
</pre_analysis>

<reasoning>

Identify the denominator and its conjugate

Using the Rationalizing the Denominator and Radical Conjugates knowledge points
\[

$$\begin{aligned} &\text{Given expression: } \frac{3}{\sqrt{17} - \sqrt{2}}\\ &\text{Denominator: } \sqrt{17} - \sqrt{2}\\ &\text{Conjugate of denominator: } \sqrt{17} + \sqrt{2} \end{aligned}$$

\]

Determine the rationalizing fraction

Using the Rationalizing the Denominator knowledge point
\[

$$\begin{aligned} &\text{To rationalize, multiply numerator and denominator by the conjugate:}\\ &\text{Fraction to multiply: } \frac{\sqrt{17} + \sqrt{2}}{\sqrt{17} + \sqrt{2}} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{\sqrt{17} - \sqrt{2}}{\sqrt{17} - \sqrt{2}}\)</mcq-option>
<mcq-correct>(B) \(\frac{\sqrt{17} + \sqrt{2}}{\sqrt{17} + \sqrt{2}}\)</mcq-correct>
<mcq-option>(C) \(\frac{\sqrt{2} - \sqrt{17}}{\sqrt{2} - \sqrt{17}}\)</mcq-option>
<mcq-option>(D) \(\frac{\sqrt{15}}{\sqrt{15}}\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Rationalizing the Denominator"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Rationalizing the Denominator",
"Radical Conjugates"
],
"new_concepts": [],
"current_concepts": [
"Rationalizing the Denominator",
"Radical Conjugates"
]
}
</pre_analysis>

<reasoning>

Identify the denominator and its conjugate

Using the Rationalizing the Denominator and Radical Conjugates knowledge points
\[

$$\begin{aligned} &\text{Given expression: } \frac{3}{\sqrt{17} - \sqrt{2}}\\ &\text{Denominator: } \sqrt{17} - \sqrt{2}\\ &\text{Conjugate of denominator: } \sqrt{17} + \sqrt{2} \end{aligned}$$

\]

Determine the rationalizing fraction

Using the Rationalizing the Denominator knowledge point
\[

$$\begin{aligned} &\text{To rationalize, multiply numerator and denominator by the conjugate:}\\ &\text{Fraction to multiply: } \frac{\sqrt{17} + \sqrt{2}}{\sqrt{17} + \sqrt{2}} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{\sqrt{17} - \sqrt{2}}{\sqrt{17} - \sqrt{2}}\)</mcq-option>
<mcq-correct>(B) \(\frac{\sqrt{17} + \sqrt{2}}{\sqrt{17} + \sqrt{2}}\)</mcq-correct>
<mcq-option>(C) \(\frac{\sqrt{2} - \sqrt{17}}{\sqrt{2} - \sqrt{17}}\)</mcq-option>
<mcq-option>(D) \(\frac{\sqrt{15}}{\sqrt{15}}\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Rationalizing the Denominator"
]
}
</post_analysis>