Algebra Examples

$v (R) = \sqrt{\frac{2 G M}{R}}$

Step 1

Write $v (R) = \sqrt{\frac{2 G M}{R}}$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt{\frac{2 y M}{R}} = G$ .

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

Step 3.2

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

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

${({(\frac{2 y M}{R})}^{\frac{1}{2}})}^{2} = G^{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${({(\frac{2 y M}{R})}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${({(\frac{2 y M}{R})}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1.1

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

${(\frac{2 y M}{R})}^{\frac{1}{2} \cdot 2} = G^{2}$

Step 3.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.2.1

Cancel the common factor.

${(\frac{2 y M}{R})}^{\frac{1}{2} \cdot 2} = G^{2}$

Step 3.3.2.1.1.2.2

Rewrite the expression.

${(\frac{2 y M}{R})}^{1} = G^{2}$

Step 3.3.2.1.2

Simplify.

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

Step 3.4

Solve for $y$ .

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Step 3.4.1

Multiply both sides by $R$ .

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $R$ .

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Step 3.4.2.1.1

Cancel the common factor.

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

Step 3.4.2.1.2

Rewrite the expression.

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

Step 3.4.3

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

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Step 3.4.3.1

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

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

Step 3.4.3.2

Simplify the left side.

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Step 3.4.3.2.1

Cancel the common factor of $2$ .

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Step 3.4.3.2.1.1

Cancel the common factor.

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

Step 3.4.3.2.1.2

Rewrite the expression.

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

Step 3.4.3.2.2

Cancel the common factor of $M$ .

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Step 3.4.3.2.2.1

Cancel the common factor.

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

Step 3.4.3.2.2.2

Divide $y$ by $1$ .

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

Step 4

Replace $y$ with $v^{- 1} (G)$ to show the final answer.

$v^{- 1} (G) = \frac{G^{2} R}{2 M}$

Step 5

Verify if $v^{- 1} (G) = \frac{G^{2} R}{2 M}$ is the inverse of $v (G) = \sqrt{\frac{2 G M}{R}}$ .

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Step 5.1

To verify the inverse, check if $v^{- 1} (v (G)) = G$ and $v (v^{- 1} (G)) = G$ .

Step 5.2

Evaluate $v^{- 1} (v (G))$ .

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Step 5.2.1

Set up the composite result function.

$v^{- 1} (v (G))$

Step 5.2.2

Evaluate $v^{- 1} (\sqrt{\frac{2 G M}{R}})$ by substituting in the value of $v$ into $v^{- 1}$ .

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{{(\sqrt{\frac{2 G M}{R}})}^{2} R}{2 M}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

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

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{{(\frac{\sqrt{2 G M}}{\sqrt{R}})}^{2} R}{2 M}$

Step 5.2.3.2

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

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{{(\frac{\sqrt{2 G M}}{\sqrt{R}} \cdot \frac{\sqrt{R}}{\sqrt{R}})}^{2} R}{2 M}$

Step 5.2.3.3

Combine and simplify the denominator.

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Step 5.2.3.3.1

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

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{{(\frac{\sqrt{2 G M} \sqrt{R}}{\sqrt{R} \sqrt{R}})}^{2} R}{2 M}$

Step 5.2.3.3.2

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

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{{(\frac{\sqrt{2 G M} \sqrt{R}}{\sqrt{R} \sqrt{R}})}^{2} R}{2 M}$

Step 5.2.3.3.3

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

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{{(\frac{\sqrt{2 G M} \sqrt{R}}{\sqrt{R} \sqrt{R}})}^{2} R}{2 M}$

Step 5.2.3.3.4

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

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{{(\frac{\sqrt{2 G M} \sqrt{R}}{{\sqrt{R}}^{1 + 1}})}^{2} R}{2 M}$

Step 5.2.3.3.5

Add $1$ and $1$ .

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{{(\frac{\sqrt{2 G M} \sqrt{R}}{{\sqrt{R}}^{2}})}^{2} R}{2 M}$

Step 5.2.3.3.6

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

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Step 5.2.3.3.6.1

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

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{{(\frac{\sqrt{2 G M} \sqrt{R}}{{(R^{\frac{1}{2}})}^{2}})}^{2} R}{2 M}$

Step 5.2.3.3.6.2

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

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{{(\frac{\sqrt{2 G M} \sqrt{R}}{R^{\frac{1}{2} \cdot 2}})}^{2} R}{2 M}$

Step 5.2.3.3.6.3

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

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{{(\frac{\sqrt{2 G M} \sqrt{R}}{R^{\frac{2}{2}}})}^{2} R}{2 M}$

Step 5.2.3.3.6.4

Cancel the common factor of $2$ .

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Step 5.2.3.3.6.4.1

Cancel the common factor.

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{{(\frac{\sqrt{2 G M} \sqrt{R}}{R^{\frac{2}{2}}})}^{2} R}{2 M}$

Step 5.2.3.3.6.4.2

Rewrite the expression.

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{{(\frac{\sqrt{2 G M} \sqrt{R}}{R})}^{2} R}{2 M}$

Step 5.2.3.3.6.5

Simplify.

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{{(\frac{\sqrt{2 G M} \sqrt{R}}{R})}^{2} R}{2 M}$

Step 5.2.3.4

Combine using the product rule for radicals.

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{{(\frac{\sqrt{2 G M R}}{R})}^{2} R}{2 M}$

Step 5.2.3.5

Apply the product rule to $\frac{\sqrt{2 G M R}}{R}$ .

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{\frac{{\sqrt{2 G M R}}^{2}}{R^{2}} \cdot R}{2 M}$

Step 5.2.3.6

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

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Step 5.2.3.6.1

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

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{\frac{{({(2 G M R)}^{\frac{1}{2}})}^{2}}{R^{2}} \cdot R}{2 M}$

Step 5.2.3.6.2

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

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{\frac{{(2 G M R)}^{\frac{1}{2} \cdot 2}}{R^{2}} \cdot R}{2 M}$

Step 5.2.3.6.3

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

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{\frac{{(2 G M R)}^{\frac{2}{2}}}{R^{2}} \cdot R}{2 M}$

Step 5.2.3.6.4

Cancel the common factor of $2$ .

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Step 5.2.3.6.4.1

Cancel the common factor.

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{\frac{{(2 G M R)}^{\frac{2}{2}}}{R^{2}} \cdot R}{2 M}$

Step 5.2.3.6.4.2

Rewrite the expression.

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{\frac{2 G M R}{R^{2}} \cdot R}{2 M}$

Step 5.2.3.6.5

Simplify.

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{\frac{2 G M R}{R^{2}} \cdot R}{2 M}$

Step 5.2.3.7

Cancel the common factor of $R$ and $R^{2}$ .

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Step 5.2.3.7.1

Factor $R$ out of $2 G M R$ .

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{\frac{R (2 G M)}{R^{2}} \cdot R}{2 M}$

Step 5.2.3.7.2

Cancel the common factors.

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Step 5.2.3.7.2.1

Factor $R$ out of $R^{2}$ .

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{\frac{R (2 G M)}{R \cdot R} \cdot R}{2 M}$

Step 5.2.3.7.2.2

Cancel the common factor.

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{\frac{R (2 G M)}{R \cdot R} \cdot R}{2 M}$

Step 5.2.3.7.2.3

Rewrite the expression.

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{\frac{2 G M}{R} \cdot R}{2 M}$

Step 5.2.4

Combine $\frac{2 G M}{R}$ and $R$ .

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{\frac{2 G M R}{R}}{2 M}$

Step 5.2.5

Reduce the expression by cancelling the common factors.

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Step 5.2.5.1

Reduce the expression $\frac{2 G M R}{R}$ by cancelling the common factors.

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Step 5.2.5.1.1

Cancel the common factor.

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{\frac{2 G M R}{R}}{2 M}$

Step 5.2.5.1.2

Rewrite the expression.

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{\frac{2 G M}{1}}{2 M}$

Step 5.2.5.2

Divide $2 G M$ by $1$ .

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{2 G M}{2 M}$

Step 5.2.6

Cancel the common factor of $2$ .

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Step 5.2.6.1

Cancel the common factor.

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{2 G M}{2 M}$

Step 5.2.6.2

Rewrite the expression.

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{G M}{M}$

Step 5.2.7

Cancel the common factor of $M$ .

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Step 5.2.7.1

Cancel the common factor.

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = \frac{G M}{M}$

Step 5.2.7.2

Divide $G$ by $1$ .

$v^{- 1} (\sqrt{\frac{2 G M}{R}}) = G$

Step 5.3

Evaluate $v (v^{- 1} (G))$ .

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Step 5.3.1

Set up the composite result function.

$v (v^{- 1} (G))$

Step 5.3.2

Evaluate $v (\frac{G^{2} R}{2 M})$ by substituting in the value of $v^{- 1}$ into $v$ .

$v (\frac{G^{2} R}{2 M}) = \sqrt{2 (\frac{G^{2} R}{2 M}) \cdot \frac{M}{R}}$

Step 5.3.3

Combine $2$ and $\frac{G^{2} R}{2 M}$ .

$v (\frac{G^{2} R}{2 M}) = \sqrt{\frac{2 (G^{2} R)}{2 M} \cdot \frac{M}{R}}$

Step 5.3.4

Multiply $\frac{2 (G^{2} R)}{2 M}$ by $\frac{M}{R}$ .

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

Step 5.3.5

Reduce the expression $\frac{2 G^{2} R M}{2 M R}$ by cancelling the common factors.

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Step 5.3.5.1

Cancel the common factor.

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

Step 5.3.5.2

Rewrite the expression.

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

Step 5.3.6

Cancel the common factor of $R$ .

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Step 5.3.6.1

Cancel the common factor.

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

Step 5.3.6.2

Rewrite the expression.

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

Step 5.3.7

Cancel the common factor of $M$ .

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Step 5.3.7.1

Cancel the common factor.

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

Step 5.3.7.2

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

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

Step 5.3.8

Pull terms out from under the radical, assuming positive real numbers.

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

Step 5.4

Since $v^{- 1} (v (G)) = G$ and $v (v^{- 1} (G)) = G$ , then $v^{- 1} (G) = \frac{G^{2} R}{2 M}$ is the inverse of $v (G) = \sqrt{\frac{2 G M}{R}}$ .

$v^{- 1} (G) = \frac{G^{2} R}{2 M}$