Algebra Examples

$m = \frac{m_{0}}{\sqrt{1 - (\frac{v^{2}}{c^{2}})}}$

Step 1

Rewrite the equation as $\frac{m_{0}}{\sqrt{1 - (\frac{v^{2}}{c^{2}})}} = m$ .

$\frac{m_{0}}{\sqrt{1 - (\frac{v^{2}}{c^{2}})}} = m$

Step 2

Cross multiply.

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Step 2.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

$m \cdot (\sqrt{1 - (\frac{v^{2}}{c^{2}})}) = m_{0}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $m \cdot (\sqrt{1 - (\frac{v^{2}}{c^{2}})})$ .

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Step 2.2.1.1

Write the expression using exponents.

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Step 2.2.1.1.1

Rewrite $1$ as $1^{2}$ .

$m \cdot \sqrt{1^{2} - \frac{v^{2}}{c^{2}}} = m_{0}$

Step 2.2.1.1.2

Rewrite $\frac{v^{2}}{c^{2}}$ as ${(\frac{v}{c})}^{2}$ .

$m \cdot \sqrt{1^{2} - {(\frac{v}{c})}^{2}} = m_{0}$

Step 2.2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{v}{c}$ .

$m \cdot \sqrt{(1 + \frac{v}{c}) (1 - \frac{v}{c})} = m_{0}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

Write $1$ as a fraction with a common denominator.

$m \cdot \sqrt{(\frac{c}{c} + \frac{v}{c}) (1 - \frac{v}{c})} = m_{0}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$m \cdot \sqrt{\frac{c + v}{c} (1 - \frac{v}{c})} = m_{0}$

Step 2.2.1.3.3

Write $1$ as a fraction with a common denominator.

$m \cdot \sqrt{\frac{c + v}{c} (\frac{c}{c} - \frac{v}{c})} = m_{0}$

Step 2.2.1.3.4

Combine the numerators over the common denominator.

$m \cdot \sqrt{\frac{c + v}{c} \cdot \frac{c - v}{c}} = m_{0}$

Step 2.2.1.3.5

Multiply $\frac{c + v}{c}$ by $\frac{c - v}{c}$ .

$m \cdot \sqrt{\frac{(c + v) (c - v)}{c \cdot c}} = m_{0}$

Step 2.2.1.3.6

Multiply $c$ by $c$ .

$m \cdot \sqrt{\frac{(c + v) (c - v)}{c^{2}}} = m_{0}$

Step 2.2.1.4

Rewrite $\frac{(c + v) (c - v)}{c^{2}}$ as ${(\frac{1}{c})}^{2} ((c + v) (c - v))$ .

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Step 2.2.1.4.1

Factor the perfect power $1^{2}$ out of $(c + v) (c - v)$ .

$m \cdot \sqrt{\frac{1^{2} ((c + v) (c - v))}{c^{2}}} = m_{0}$

Step 2.2.1.4.2

Factor the perfect power $c^{2}$ out of $c^{2}$ .

$m \cdot \sqrt{\frac{1^{2} ((c + v) (c - v))}{c^{2} \cdot 1}} = m_{0}$

Step 2.2.1.4.3

Rearrange the fraction $\frac{1^{2} ((c + v) (c - v))}{c^{2} \cdot 1}$ .

$m \cdot \sqrt{{(\frac{1}{c})}^{2} ((c + v) (c - v))} = m_{0}$

Step 2.2.1.5

Pull terms out from under the radical.

$m \cdot (\frac{1}{c} \sqrt{(c + v) (c - v)}) = m_{0}$

Step 2.2.1.6

Combine $\frac{1}{c}$ and $\sqrt{(c + v) (c - v)}$ .

$\frac{m \sqrt{(c + v) (c - v)}}{c} = m_{0}$

Step 3

Solve for $m \sqrt{(c + v) (c - v)}$ .

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Step 3.1

Multiply both sides by $c$ .

$\frac{m \sqrt{(c + v) (c - v)}}{c} c = m_{0} c$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $c$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{m \sqrt{(c + v) (c - v)}}{c} c = m_{0} c$

Step 3.2.1.2

Rewrite the expression.

$m \sqrt{(c + v) (c - v)} = m_{0} c$

Step 4

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

${(m \sqrt{(c + v) (c - v)})}^{2} = {(m_{0} c)}^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(c + v) (c - v)}$ as ${((c + v) (c - v))}^{\frac{1}{2}}$ .

${(m {((c + v) (c - v))}^{\frac{1}{2}})}^{2} = {(m_{0} c)}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify ${(m {((c + v) (c - v))}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1

Expand $(c + v) (c - v)$ using the FOIL Method.

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Step 5.2.1.1.1

Apply the distributive property.

${(m {(c (c - v) + v (c - v))}^{\frac{1}{2}})}^{2} = {(m_{0} c)}^{2}$

Step 5.2.1.1.2

Apply the distributive property.

${(m {(c \cdot c + c (- v) + v (c - v))}^{\frac{1}{2}})}^{2} = {(m_{0} c)}^{2}$

Step 5.2.1.1.3

Apply the distributive property.

${(m {(c \cdot c + c (- v) + v c + v (- v))}^{\frac{1}{2}})}^{2} = {(m_{0} c)}^{2}$

Step 5.2.1.2

Simplify terms.

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Step 5.2.1.2.1

Combine the opposite terms in $c \cdot c + c (- v) + v c + v (- v)$ .

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Step 5.2.1.2.1.1

Reorder the factors in the terms $c (- v)$ and $v c$ .

${(m {(c \cdot c - c v + c v + v (- v))}^{\frac{1}{2}})}^{2} = {(m_{0} c)}^{2}$

Step 5.2.1.2.1.2

Add $- c v$ and $c v$ .

${(m {(c \cdot c + 0 + v (- v))}^{\frac{1}{2}})}^{2} = {(m_{0} c)}^{2}$

Step 5.2.1.2.1.3

Add $c \cdot c$ and $0$ .

${(m {(c \cdot c + v (- v))}^{\frac{1}{2}})}^{2} = {(m_{0} c)}^{2}$

Step 5.2.1.2.2

Simplify each term.

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Step 5.2.1.2.2.1

Multiply $c$ by $c$ .

${(m {(c^{2} + v (- v))}^{\frac{1}{2}})}^{2} = {(m_{0} c)}^{2}$

Step 5.2.1.2.2.2

Rewrite using the commutative property of multiplication.

${(m {(c^{2} - v \cdot v)}^{\frac{1}{2}})}^{2} = {(m_{0} c)}^{2}$

Step 5.2.1.2.2.3

Multiply $v$ by $v$ by adding the exponents.

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Step 5.2.1.2.2.3.1

Move $v$ .

${(m {(c^{2} - (v \cdot v))}^{\frac{1}{2}})}^{2} = {(m_{0} c)}^{2}$

Step 5.2.1.2.2.3.2

Multiply $v$ by $v$ .

${(m {(c^{2} - v^{2})}^{\frac{1}{2}})}^{2} = {(m_{0} c)}^{2}$

Step 5.2.1.2.3

Apply basic rules of exponents.

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Step 5.2.1.2.3.1

Apply the product rule to $m {(c^{2} - v^{2})}^{\frac{1}{2}}$ .

$m^{2} {({(c^{2} - v^{2})}^{\frac{1}{2}})}^{2} = {(m_{0} c)}^{2}$

Step 5.2.1.2.3.2

Multiply the exponents in ${({(c^{2} - v^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.2.3.2.1

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

$m^{2} {(c^{2} - v^{2})}^{\frac{1}{2} \cdot 2} = {(m_{0} c)}^{2}$

Step 5.2.1.2.3.2.2

Cancel the common factor of $2$ .

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Step 5.2.1.2.3.2.2.1

Cancel the common factor.

$m^{2} {(c^{2} - v^{2})}^{\frac{1}{2} \cdot 2} = {(m_{0} c)}^{2}$

Step 5.2.1.2.3.2.2.2

Rewrite the expression.

$m^{2} {(c^{2} - v^{2})}^{1} = {(m_{0} c)}^{2}$

Step 5.2.1.3

Simplify.

$m^{2} (c^{2} - v^{2}) = {(m_{0} c)}^{2}$

Step 5.2.1.4

Apply the distributive property.

$m^{2} c^{2} + m^{2} (- v^{2}) = {(m_{0} c)}^{2}$

Step 5.2.1.5

Rewrite using the commutative property of multiplication.

$m^{2} c^{2} - m^{2} v^{2} = {(m_{0} c)}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Apply the product rule to $m_{0} c$ .

$m^{2} c^{2} - m^{2} v^{2} = {m_{0}}^{2} c^{2}$

Step 6

Solve for $v$ .

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Step 6.1

Subtract $m^{2} c^{2}$ from both sides of the equation.

$- m^{2} v^{2} = {m_{0}}^{2} c^{2} - m^{2} c^{2}$

Step 6.2

Divide each term in $- m^{2} v^{2} = {m_{0}}^{2} c^{2} - m^{2} c^{2}$ by $- m^{2}$ and simplify.

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Step 6.2.1

Divide each term in $- m^{2} v^{2} = {m_{0}}^{2} c^{2} - m^{2} c^{2}$ by $- m^{2}$ .

$\frac{- m^{2} v^{2}}{- m^{2}} = \frac{{m_{0}}^{2} c^{2}}{- m^{2}} + \frac{- m^{2} c^{2}}{- m^{2}}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Dividing two negative values results in a positive value.

$\frac{m^{2} v^{2}}{m^{2}} = \frac{{m_{0}}^{2} c^{2}}{- m^{2}} + \frac{- m^{2} c^{2}}{- m^{2}}$

Step 6.2.2.2

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

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Step 6.2.2.2.1

Cancel the common factor.

$\frac{m^{2} v^{2}}{m^{2}} = \frac{{m_{0}}^{2} c^{2}}{- m^{2}} + \frac{- m^{2} c^{2}}{- m^{2}}$

Step 6.2.2.2.2

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

$v^{2} = \frac{{m_{0}}^{2} c^{2}}{- m^{2}} + \frac{- m^{2} c^{2}}{- m^{2}}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Simplify each term.

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Step 6.2.3.1.1

Move the negative in front of the fraction.

$v^{2} = - \frac{{m_{0}}^{2} c^{2}}{m^{2}} + \frac{- m^{2} c^{2}}{- m^{2}}$

Step 6.2.3.1.2

Dividing two negative values results in a positive value.

$v^{2} = - \frac{{m_{0}}^{2} c^{2}}{m^{2}} + \frac{m^{2} c^{2}}{m^{2}}$

Step 6.2.3.1.3

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

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Step 6.2.3.1.3.1

Cancel the common factor.

$v^{2} = - \frac{{m_{0}}^{2} c^{2}}{m^{2}} + \frac{m^{2} c^{2}}{m^{2}}$

Step 6.2.3.1.3.2

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

$v^{2} = - \frac{{m_{0}}^{2} c^{2}}{m^{2}} + c^{2}$

Step 6.3

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

$v = \pm \sqrt{- \frac{{m_{0}}^{2} c^{2}}{m^{2}} + c^{2}}$

Step 6.4

Simplify $\pm \sqrt{- \frac{{m_{0}}^{2} c^{2}}{m^{2}} + c^{2}}$ .

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Step 6.4.1

Factor $c^{2}$ out of $- \frac{{m_{0}}^{2} c^{2}}{m^{2}} + c^{2}$ .

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Step 6.4.1.1

Factor $c^{2}$ out of $- \frac{{m_{0}}^{2} c^{2}}{m^{2}}$ .

$v = \pm \sqrt{c^{2} (- \frac{{m_{0}}^{2}}{m^{2}}) + c^{2}}$

Step 6.4.1.2

Multiply by $1$ .

$v = \pm \sqrt{c^{2} (- \frac{{m_{0}}^{2}}{m^{2}}) + c^{2} \cdot 1}$

Step 6.4.1.3

Factor $c^{2}$ out of $c^{2} (- \frac{{m_{0}}^{2}}{m^{2}}) + c^{2} \cdot 1$ .

$v = \pm \sqrt{c^{2} (- \frac{{m_{0}}^{2}}{m^{2}} + 1)}$

Step 6.4.2

Simplify the expression.

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Step 6.4.2.1

Rewrite $1$ as $1^{2}$ .

$v = \pm \sqrt{c^{2} (- \frac{{m_{0}}^{2}}{m^{2}} + 1^{2})}$

Step 6.4.2.2

Rewrite $\frac{{m_{0}}^{2}}{m^{2}}$ as ${(\frac{m_{0}}{m})}^{2}$ .

$v = \pm \sqrt{c^{2} (- {(\frac{m_{0}}{m})}^{2} + 1^{2})}$

Step 6.4.2.3

Reorder $- {(\frac{m_{0}}{m})}^{2}$ and $1^{2}$ .

$v = \pm \sqrt{c^{2} (1^{2} - {(\frac{m_{0}}{m})}^{2})}$

Step 6.4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{m_{0}}{m}$ .

$v = \pm \sqrt{c^{2} (1 + \frac{m_{0}}{m}) (1 - \frac{m_{0}}{m})}$

Step 6.4.4

Write $1$ as a fraction with a common denominator.

$v = \pm \sqrt{c^{2} (\frac{m}{m} + \frac{m_{0}}{m}) (1 - \frac{m_{0}}{m})}$

Step 6.4.5

Combine the numerators over the common denominator.

$v = \pm \sqrt{c^{2} \frac{m + m_{0}}{m} (1 - \frac{m_{0}}{m})}$

Step 6.4.6

Write $1$ as a fraction with a common denominator.

$v = \pm \sqrt{c^{2} \frac{m + m_{0}}{m} (\frac{m}{m} - \frac{m_{0}}{m})}$

Step 6.4.7

Combine the numerators over the common denominator.

$v = \pm \sqrt{c^{2} \frac{m + m_{0}}{m} \frac{m - m_{0}}{m}}$

Step 6.4.8

Combine exponents.

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Step 6.4.8.1

Combine $c^{2}$ and $\frac{m + m_{0}}{m}$ .

$v = \pm \sqrt{\frac{c^{2} (m + m_{0})}{m} \cdot \frac{m - m_{0}}{m}}$

Step 6.4.8.2

Multiply $\frac{c^{2} (m + m_{0})}{m}$ by $\frac{m - m_{0}}{m}$ .

$v = \pm \sqrt{\frac{c^{2} (m + m_{0}) (m - m_{0})}{m \cdot m}}$

Step 6.4.8.3

Raise $m$ to the power of $1$ .

$v = \pm \sqrt{\frac{c^{2} (m + m_{0}) (m - m_{0})}{m^{1} m}}$

Step 6.4.8.4

Raise $m$ to the power of $1$ .

$v = \pm \sqrt{\frac{c^{2} (m + m_{0}) (m - m_{0})}{m^{1} m^{1}}}$

Step 6.4.8.5

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

$v = \pm \sqrt{\frac{c^{2} (m + m_{0}) (m - m_{0})}{m^{1 + 1}}}$

Step 6.4.8.6

Add $1$ and $1$ .

$v = \pm \sqrt{\frac{c^{2} (m + m_{0}) (m - m_{0})}{m^{2}}}$

Step 6.4.9

Rewrite $\frac{c^{2} (m + m_{0}) (m - m_{0})}{m^{2}}$ as ${(\frac{c}{m})}^{2} ((m + m_{0}) (m - m_{0}))$ .

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Step 6.4.9.1

Factor the perfect power $c^{2}$ out of $c^{2} (m + m_{0}) (m - m_{0})$ .

$v = \pm \sqrt{\frac{c^{2} ((m + m_{0}) (m - m_{0}))}{m^{2}}}$

Step 6.4.9.2

Factor the perfect power $m^{2}$ out of $m^{2}$ .

$v = \pm \sqrt{\frac{c^{2} ((m + m_{0}) (m - m_{0}))}{m^{2} \cdot 1}}$

Step 6.4.9.3

Rearrange the fraction $\frac{c^{2} ((m + m_{0}) (m - m_{0}))}{m^{2} \cdot 1}$ .

$v = \pm \sqrt{{(\frac{c}{m})}^{2} ((m + m_{0}) (m - m_{0}))}$

Step 6.4.10

Pull terms out from under the radical.

$v = \pm \frac{c}{m} \sqrt{(m + m_{0}) (m - m_{0})}$

Step 6.4.11

Combine $\frac{c}{m}$ and $\sqrt{(m + m_{0}) (m - m_{0})}$ .

$v = \pm \frac{c \sqrt{(m + m_{0}) (m - m_{0})}}{m}$

Step 6.5

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

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Step 6.5.1

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

$v = \frac{c \sqrt{(m + m_{0}) (m - m_{0})}}{m}$

Step 6.5.2

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

$v = - \frac{c \sqrt{(m + m_{0}) (m - m_{0})}}{m}$

Step 6.5.3

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

$v = \frac{c \sqrt{(m + m_{0}) (m - m_{0})}}{m}$

$v = - \frac{c \sqrt{(m + m_{0}) (m - m_{0})}}{m}$

$v = \frac{c \sqrt{(m + m_{0}) (m - m_{0})}}{m}$

$v = - \frac{c \sqrt{(m + m_{0}) (m - m_{0})}}{m}$

$v = \frac{c \sqrt{(m + m_{0}) (m - m_{0})}}{m}$

$v = - \frac{c \sqrt{(m + m_{0}) (m - m_{0})}}{m}$