Question

the gravitational force formula is \\(f = \frac{g m_1 m_2}{r^2}\\), where \\(f\\) is the force between two objects, \\(g\\) is the constant of gravitation, \\(m_1\\) is the mass of the first object, \\(m_2\\) is the mass of the second object, and \\(r\\) is the distance between the objects. by rewriting the formula as \\(r = \sqrt{\frac{g m_1 m_2}{f}}\\), you can find the distance between objects. which of the following gives the distance, \\(r\\), in simplest form?

\\(r = \frac{\sqrt{g m_1 m_2}}{f}\\)

\\(r = \frac{\sqrt{g m_1 m_2 f}}{f}\\)

\\(r = \sqrt{g m_1 m_2 f}\\)

Explanation:

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<reasoning>

Apply the quotient property of radicals

Using the Quotient Property of Radicals knowledge point
\[
r = \sqrt{\frac{G m_1 m_2}{F}} = \frac{\sqrt{G m_1 m_2}}{\sqrt{F}}
\]

Rationalize the denominator

Using the Rationalizing the Denominator knowledge point
\[
r = \frac{\sqrt{G m_1 m_2} \cdot \sqrt{F}}{\sqrt{F} \cdot \sqrt{F}} = \frac{\sqrt{G m_1 m_2 F}}{F}
\]
</reasoning>

<answer>
<mcq-option>(A) \(r = \frac{\sqrt{G m_1 m_2}}{F}\)</mcq-option>
<mcq-correct>(B) \(r = \frac{\sqrt{G m_1 m_2 F}}{F}\)</mcq-correct>
<mcq-option>(C) \(r = \sqrt{G m_1 m_2 F}\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Apply the quotient property of radicals

Using the Quotient Property of Radicals knowledge point
\[
r = \sqrt{\frac{G m_1 m_2}{F}} = \frac{\sqrt{G m_1 m_2}}{\sqrt{F}}
\]

Rationalize the denominator

Using the Rationalizing the Denominator knowledge point
\[
r = \frac{\sqrt{G m_1 m_2} \cdot \sqrt{F}}{\sqrt{F} \cdot \sqrt{F}} = \frac{\sqrt{G m_1 m_2 F}}{F}
\]
</reasoning>

<answer>
<mcq-option>(A) \(r = \frac{\sqrt{G m_1 m_2}}{F}\)</mcq-option>
<mcq-correct>(B) \(r = \frac{\sqrt{G m_1 m_2 F}}{F}\)</mcq-correct>
<mcq-option>(C) \(r = \sqrt{G m_1 m_2 F}\)</mcq-option>
</answer>

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