Question

the gravitational force formula is $f = \frac{gm_1m_2}{r^{2}}$, 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 $r = \sqrt{\frac{gm_1m_2}{f}}$, you can find the distance between objects. which of the following gives the distance, $r$, in simplest form? $r = \frac{\sqrt{gm_1m_2}}{f}$ $r = \sqrt{gm_1m_2f}$ $r = \frac{\sqrt{gm_1m_2f}}{f}$

Explanation:

Step1: Recall the square - root property

We have $r = \sqrt{\frac{Gm_1m_2}{F}}$. By the property of square - roots $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ ($a = Gm_1m_2$, $b = F$ and $F>0$).
$r=\frac{\sqrt{Gm_1m_2}}{\sqrt{F}}$

Step2: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and denominator by $\sqrt{F}$.
$r=\frac{\sqrt{Gm_1m_2}\times\sqrt{F}}{\sqrt{F}\times\sqrt{F}}=\frac{\sqrt{Gm_1m_2F}}{F}$

Answer:

$r = \frac{\sqrt{Gm_1m_2F}}{F}$