Question

the process for rationalizing a denominator in a variable expression is the same as in a numeric expression. heres a real-world example.

the kinetic energy of the car of a rollercoaster is given by the formula \\(k = \frac{1}{2}mv^2\\), where \\(k\\) is kinetic energy, \\(m\\) is the mass of the car, and \\(v\\) is the velocity of the car. solving this formula for \\(v\\), we get \\(v = \sqrt{\frac{2k}{m}}\\).

which formula gives the velocity of the car in simplest form?

\\(v = \frac{\sqrt{2km}}{2m}\\)
\\(v = \frac{\sqrt{2km}}{m}\\)
\\(v = \frac{\sqrt{2k}}{m}\\)

Explanation:

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<reasoning>

Identify the given formula and target

The problem asks us to find the formula for the velocity \(v\) in simplest form by rationalizing the denominator of the expression:
\[v = \sqrt{\frac{2k}{m}}\]

Apply the quotient property of radicals

Using the Quotient Property of Radicals knowledge point
\[v = \frac{\sqrt{2k}}{\sqrt{m}}\]

Rationalize the denominator

Using the Radical Simplification knowledge point
\[v = \frac{\sqrt{2k} \cdot \sqrt{m}}{\sqrt{m} \cdot \sqrt{m}} = \frac{\sqrt{2km}}{m}\]
</reasoning>

<answer>
<mcq-option>(A) \(v = \frac{\sqrt{2km}}{2m}\)</mcq-option>
<mcq-correct>(B) \(v = \frac{\sqrt{2km}}{m}\)</mcq-correct>
<mcq-option>(C) \(v = \frac{\sqrt{2k}}{m}\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the given formula and target

The problem asks us to find the formula for the velocity \(v\) in simplest form by rationalizing the denominator of the expression:
\[v = \sqrt{\frac{2k}{m}}\]

Apply the quotient property of radicals

Using the Quotient Property of Radicals knowledge point
\[v = \frac{\sqrt{2k}}{\sqrt{m}}\]

Rationalize the denominator

Using the Radical Simplification knowledge point
\[v = \frac{\sqrt{2k} \cdot \sqrt{m}}{\sqrt{m} \cdot \sqrt{m}} = \frac{\sqrt{2km}}{m}\]
</reasoning>

<answer>
<mcq-option>(A) \(v = \frac{\sqrt{2km}}{2m}\)</mcq-option>
<mcq-correct>(B) \(v = \frac{\sqrt{2km}}{m}\)</mcq-correct>
<mcq-option>(C) \(v = \frac{\sqrt{2k}}{m}\)</mcq-option>
</answer>

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