Solve for r r = square root of (Gm_1m_2)/F

$r = \sqrt{\frac{{Gm}_{1} m_{2}}{F}}$

Rewrite $\sqrt{\frac{G m_{1} m_{2}}{F}}$ as $\frac{\sqrt{G m_{1} m_{2}}}{\sqrt{F}}$ .

$r = \frac{\sqrt{G m_{1} m_{2}}}{\sqrt{F}}$

Multiply $\frac{\sqrt{G m_{1} m_{2}}}{\sqrt{F}}$ by $\frac{\sqrt{F}}{\sqrt{F}}$ .

$r = \frac{\sqrt{G m_{1} m_{2}}}{\sqrt{F}} \cdot \frac{\sqrt{F}}{\sqrt{F}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{G m_{1} m_{2}}}{\sqrt{F}}$ by $\frac{\sqrt{F}}{\sqrt{F}}$ .

$r = \frac{\sqrt{G m_{1} m_{2}} \sqrt{F}}{\sqrt{F} \sqrt{F}}$

Raise $\sqrt{F}$ to the power of $1$ .

$r = \frac{\sqrt{G m_{1} m_{2}} \sqrt{F}}{{\sqrt{F}}^{1} \sqrt{F}}$

Raise $\sqrt{F}$ to the power of $1$ .

$r = \frac{\sqrt{G m_{1} m_{2}} \sqrt{F}}{{\sqrt{F}}^{1} {\sqrt{F}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$r = \frac{\sqrt{G m_{1} m_{2}} \sqrt{F}}{{\sqrt{F}}^{1 + 1}}$

Add $1$ and $1$ .

$r = \frac{\sqrt{G m_{1} m_{2}} \sqrt{F}}{{\sqrt{F}}^{2}}$

Rewrite ${\sqrt{F}}^{2}$ as $F$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{F}$ as $F^{\frac{1}{2}}$ .

$r = \frac{\sqrt{G m_{1} m_{2}} \sqrt{F}}{{(F^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$r = \frac{\sqrt{G m_{1} m_{2}} \sqrt{F}}{F^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$r = \frac{\sqrt{G m_{1} m_{2}} \sqrt{F}}{F^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$r = \frac{\sqrt{G m_{1} m_{2}} \sqrt{F}}{F^{\frac{2}{2}}}$

Rewrite the expression.

$r = \frac{\sqrt{G m_{1} m_{2}} \sqrt{F}}{F^{1}}$

Simplify.

$r = \frac{\sqrt{G m_{1} m_{2}} \sqrt{F}}{F}$

Combine using the product rule for radicals.

$r = \frac{\sqrt{G m_{1} m_{2} F}}{F}$