Solve for r Fg=(GM_1M_2)/(r^2)

$F g = \frac{G M_{1} M_{2}}{r^{2}}$

Step 1

Rewrite the equation as $\frac{G M_{1} M_{2}}{r^{2}} = F g$ .

$\frac{G M_{1} M_{2}}{r^{2}} = F g$

Step 2

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$r^{2}, 1$

The LCM of one and any expression is the expression.

$r^{2}$

Step 3

Multiply each term in $\frac{G M_{1} M_{2}}{r^{2}} = F g$ by $r^{2}$ to eliminate the fractions.

Multiply each term in $\frac{G M_{1} M_{2}}{r^{2}} = F g$ by $r^{2}$ .

$\frac{G M_{1} M_{2}}{r^{2}} r^{2} = F g r^{2}$

Simplify the left side.

Cancel the common factor of $r^{2}$ .

Cancel the common factor.

$\frac{G M_{1} M_{2}}{r^{2}} r^{2} = F g r^{2}$

Rewrite the expression.

$G M_{1} M_{2} = F g r^{2}$

Step 4

Solve the equation.

Rewrite the equation as $F g r^{2} = G M_{1} M_{2}$ .

$F g r^{2} = G M_{1} M_{2}$

Divide each term in $F g r^{2} = G M_{1} M_{2}$ by $F g$ and simplify.

Divide each term in $F g r^{2} = G M_{1} M_{2}$ by $F g$ .

$\frac{F g r^{2}}{F g} = \frac{G M_{1} M_{2}}{F g}$

Simplify the left side.

Cancel the common factor of $F$ .

Cancel the common factor.

$\frac{F g r^{2}}{F g} = \frac{G M_{1} M_{2}}{F g}$

Rewrite the expression.

$\frac{g r^{2}}{g} = \frac{G M_{1} M_{2}}{F g}$

Cancel the common factor of $g$ .

Cancel the common factor.

$\frac{g r^{2}}{g} = \frac{G M_{1} M_{2}}{F g}$

Divide $r^{2}$ by $1$ .

$r^{2} = \frac{G M_{1} M_{2}}{F g}$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$r = \pm \sqrt{\frac{G M_{1} M_{2}}{F g}}$

Simplify $\pm \sqrt{\frac{G M_{1} M_{2}}{F g}}$ .

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

$r = \pm \frac{\sqrt{G M_{1} M_{2}}}{\sqrt{F g}}$

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

$r = \pm \frac{\sqrt{G M_{1} M_{2}}}{\sqrt{F g}} \cdot \frac{\sqrt{F g}}{\sqrt{F g}}$

Combine and simplify the denominator.

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

$r = \pm \frac{\sqrt{G M_{1} M_{2}} \sqrt{F g}}{\sqrt{F g} \sqrt{F g}}$

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

$r = \pm \frac{\sqrt{G M_{1} M_{2}} \sqrt{F g}}{{\sqrt{F g}}^{1} \sqrt{F g}}$

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

$r = \pm \frac{\sqrt{G M_{1} M_{2}} \sqrt{F g}}{{\sqrt{F g}}^{1} {\sqrt{F g}}^{1}}$

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

$r = \pm \frac{\sqrt{G M_{1} M_{2}} \sqrt{F g}}{{\sqrt{F g}}^{1 + 1}}$

Add $1$ and $1$ .

$r = \pm \frac{\sqrt{G M_{1} M_{2}} \sqrt{F g}}{{\sqrt{F g}}^{2}}$

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

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

$r = \pm \frac{\sqrt{G M_{1} M_{2}} \sqrt{F g}}{{({(F g)}^{\frac{1}{2}})}^{2}}$

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

$r = \pm \frac{\sqrt{G M_{1} M_{2}} \sqrt{F g}}{{(F g)}^{\frac{1}{2} \cdot 2}}$

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

$r = \pm \frac{\sqrt{G M_{1} M_{2}} \sqrt{F g}}{{(F g)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$r = \pm \frac{\sqrt{G M_{1} M_{2}} \sqrt{F g}}{{(F g)}^{\frac{2}{2}}}$

Rewrite the expression.

$r = \pm \frac{\sqrt{G M_{1} M_{2}} \sqrt{F g}}{{(F g)}^{1}}$

Simplify.

$r = \pm \frac{\sqrt{G M_{1} M_{2}} \sqrt{F g}}{F g}$

Combine using the product rule for radicals.

$r = \pm \frac{\sqrt{G M_{1} M_{2} F g}}{F g}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

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

Next, use the negative value of the $\pm$ to find the second solution.

$r = - \frac{\sqrt{G M_{1} M_{2} F g}}{F g}$

The complete solution is the result of both the positive and negative portions of the solution.

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

$r = - \frac{\sqrt{G M_{1} M_{2} F g}}{F g}$

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

$r = - \frac{\sqrt{G M_{1} M_{2} F g}}{F g}$

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

$r = - \frac{\sqrt{G M_{1} M_{2} F g}}{F g}$

Solve for r Fg=(GM_1M_2)/(r^2)

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