Finite Math Examples

$R = \frac{2 (G M)}{c^{2}}$

Step 1

Rewrite the equation as $\frac{2 G M}{c^{2}} = R$ .

$\frac{2 G M}{c^{2}} = R$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$c^{2}, 1$

Step 2.2

The LCM of one and any expression is the expression.

$c^{2}$

Step 3

Multiply each term in $\frac{2 G M}{c^{2}} = R$ by $c^{2}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{2 G M}{c^{2}} = R$ by $c^{2}$ .

$\frac{2 G M}{c^{2}} c^{2} = R c^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 G M}{c^{2}} c^{2} = R c^{2}$

Step 3.2.1.2

Rewrite the expression.

$2 G M = R c^{2}$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $R c^{2} = 2 G M$ .

$R c^{2} = 2 G M$

Step 4.2

Divide each term in $R c^{2} = 2 G M$ by $R$ and simplify.

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Step 4.2.1

Divide each term in $R c^{2} = 2 G M$ by $R$ .

$\frac{R c^{2}}{R} = \frac{2 G M}{R}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $R$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{R c^{2}}{R} = \frac{2 G M}{R}$

Step 4.2.2.1.2

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

$c^{2} = \frac{2 G M}{R}$

Step 4.3

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

$c = \pm \sqrt{\frac{2 G M}{R}}$

Step 4.4

Simplify $\pm \sqrt{\frac{2 G M}{R}}$ .

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Step 4.4.1

Rewrite $\sqrt{\frac{2 G M}{R}}$ as $\frac{\sqrt{2 G M}}{\sqrt{R}}$ .

$c = \pm \frac{\sqrt{2 G M}}{\sqrt{R}}$

Step 4.4.2

Multiply $\frac{\sqrt{2 G M}}{\sqrt{R}}$ by $\frac{\sqrt{R}}{\sqrt{R}}$ .

$c = \pm \frac{\sqrt{2 G M}}{\sqrt{R}} \cdot \frac{\sqrt{R}}{\sqrt{R}}$

Step 4.4.3

Combine and simplify the denominator.

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Step 4.4.3.1

Multiply $\frac{\sqrt{2 G M}}{\sqrt{R}}$ by $\frac{\sqrt{R}}{\sqrt{R}}$ .

$c = \pm \frac{\sqrt{2 G M} \sqrt{R}}{\sqrt{R} \sqrt{R}}$

Step 4.4.3.2

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

$c = \pm \frac{\sqrt{2 G M} \sqrt{R}}{{\sqrt{R}}^{1} \sqrt{R}}$

Step 4.4.3.3

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

$c = \pm \frac{\sqrt{2 G M} \sqrt{R}}{{\sqrt{R}}^{1} {\sqrt{R}}^{1}}$

Step 4.4.3.4

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

$c = \pm \frac{\sqrt{2 G M} \sqrt{R}}{{\sqrt{R}}^{1 + 1}}$

Step 4.4.3.5

Add $1$ and $1$ .

$c = \pm \frac{\sqrt{2 G M} \sqrt{R}}{{\sqrt{R}}^{2}}$

Step 4.4.3.6

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

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Step 4.4.3.6.1

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

$c = \pm \frac{\sqrt{2 G M} \sqrt{R}}{{(R^{\frac{1}{2}})}^{2}}$

Step 4.4.3.6.2

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

$c = \pm \frac{\sqrt{2 G M} \sqrt{R}}{R^{\frac{1}{2} \cdot 2}}$

Step 4.4.3.6.3

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

$c = \pm \frac{\sqrt{2 G M} \sqrt{R}}{R^{\frac{2}{2}}}$

Step 4.4.3.6.4

Cancel the common factor of $2$ .

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Step 4.4.3.6.4.1

Cancel the common factor.

$c = \pm \frac{\sqrt{2 G M} \sqrt{R}}{R^{\frac{2}{2}}}$

Step 4.4.3.6.4.2

Rewrite the expression.

$c = \pm \frac{\sqrt{2 G M} \sqrt{R}}{R^{1}}$

Step 4.4.3.6.5

Simplify.

$c = \pm \frac{\sqrt{2 G M} \sqrt{R}}{R}$

Step 4.4.4

Combine using the product rule for radicals.

$c = \pm \frac{\sqrt{2 G M R}}{R}$

Step 4.5

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

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Step 4.5.1

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

$c = \frac{\sqrt{2 G M R}}{R}$

Step 4.5.2

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

$c = - \frac{\sqrt{2 G M R}}{R}$

Step 4.5.3

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

$c = \frac{\sqrt{2 G M R}}{R}$

$c = - \frac{\sqrt{2 G M R}}{R}$

$c = \frac{\sqrt{2 G M R}}{R}$

$c = - \frac{\sqrt{2 G M R}}{R}$

$c = \frac{\sqrt{2 G M R}}{R}$

$c = - \frac{\sqrt{2 G M R}}{R}$