Solve for r F_g=G(m_1m_2)/(r^2)

$F_{g} = G \frac{m_{1} m_{2}}{r^{2}}$

Step 1

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

$G \frac{m_{1} m_{2}}{r^{2}} = F_{g}$

Step 2

Combine $G$ and $\frac{m_{1} m_{2}}{r^{2}}$ .

$\frac{G m_{1} m_{2}}{r^{2}} = F_{g}$

Step 3

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 4

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 5

Solve the equation.

Rewrite the equation as $F_{g} r^{2} = GCF$ .

$F_{g} r^{2} = GCF$

Divide each term in $F_{g} r^{2} = GCF$ by $F_{g}$ and simplify.

Divide each term in $F_{g} r^{2} = GCF$ by $F_{g}$ .

$\frac{F_{g} r^{2}}{F_{g}} = \frac{GCF}{F_{g}}$

Simplify the left side.

Cancel the common factor of $F_{g}$ .

Cancel the common factor.

$\frac{F_{g} r^{2}}{F_{g}} = \frac{GCF}{F_{g}}$

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

$r^{2} = \frac{GCF}{F_{g}}$

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

$r = \pm \sqrt{\frac{GCF}{F_{g}}}$

Simplify $\pm \sqrt{\frac{GCF}{F_{g}}}$ .

Rewrite $\sqrt{\frac{GCF}{F_{g}}}$ as $\frac{\sqrt{GCF}}{\sqrt{F_{g}}}$ .

$r = \pm \frac{\sqrt{GCF}}{\sqrt{F_{g}}}$

Multiply $\frac{\sqrt{GCF}}{\sqrt{F_{g}}}$ by $\frac{\sqrt{F_{g}}}{\sqrt{F_{g}}}$ .

$r = \pm \frac{\sqrt{GCF}}{\sqrt{F_{g}}} \cdot \frac{\sqrt{F_{g}}}{\sqrt{F_{g}}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{GCF}}{\sqrt{F_{g}}}$ by $\frac{\sqrt{F_{g}}}{\sqrt{F_{g}}}$ .

$r = \pm \frac{\sqrt{GCF} \sqrt{F_{g}}}{\sqrt{F_{g}} \sqrt{F_{g}}}$

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

$r = \pm \frac{\sqrt{GCF} \sqrt{F_{g}}}{{\sqrt{F_{g}}}^{1} \sqrt{F_{g}}}$

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

$r = \pm \frac{\sqrt{GCF} \sqrt{F_{g}}}{{\sqrt{F_{g}}}^{1} {\sqrt{F_{g}}}^{1}}$

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

$r = \pm \frac{\sqrt{GCF} \sqrt{F_{g}}}{{\sqrt{F_{g}}}^{1 + 1}}$

Add $1$ and $1$ .

$r = \pm \frac{\sqrt{GCF} \sqrt{F_{g}}}{{\sqrt{F_{g}}}^{2}}$

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

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

$r = \pm \frac{\sqrt{GCF} \sqrt{F_{g}}}{{({F_{g}}^{\frac{1}{2}})}^{2}}$

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

$r = \pm \frac{\sqrt{GCF} \sqrt{F_{g}}}{{F_{g}}^{\frac{1}{2} \cdot 2}}$

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

$r = \pm \frac{\sqrt{GCF} \sqrt{F_{g}}}{{F_{g}}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$r = \pm \frac{\sqrt{GCF} \sqrt{F_{g}}}{{F_{g}}^{\frac{2}{2}}}$

Rewrite the expression.

$r = \pm \frac{\sqrt{GCF} \sqrt{F_{g}}}{{F_{g}}^{1}}$

Simplify.

$r = \pm \frac{\sqrt{GCF} \sqrt{F_{g}}}{F_{g}}$

Combine using the product rule for radicals.

$r = \pm \frac{\sqrt{GCF F_{g}}}{F_{g}}$

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.

$r = \frac{\sqrt{GCF F_{g}}}{F_{g}}$

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

$r = - \frac{\sqrt{GCF F_{g}}}{F_{g}}$

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

$r = \frac{\sqrt{GCF F_{g}}}{F_{g}}$

$r = - \frac{\sqrt{GCF F_{g}}}{F_{g}}$

$r = \frac{\sqrt{GCF F_{g}}}{F_{g}}$

$r = - \frac{\sqrt{GCF F_{g}}}{F_{g}}$

$r = \frac{\sqrt{GCF F_{g}}}{F_{g}}$

$r = - \frac{\sqrt{GCF F_{g}}}{F_{g}}$

Solve for r F_g=G(m_1m_2)/(r^2)

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