Algebra Examples

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

Step 1

Multiply $- 1$ by $\frac{v^{2}}{c^{2}}$ .

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

Step 2

Remove parentheses.

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

Step 3

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

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Step 3.1

Simplify the denominator.

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Step 3.1.1

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

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

Step 3.1.2

Combine the numerators over the common denominator.

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

Step 3.1.3

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

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

Step 3.1.4

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

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Step 3.1.4.1

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

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

Step 3.1.4.2

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

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

Step 3.1.4.3

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

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

Step 3.1.5

Pull terms out from under the radical.

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

Step 3.1.6

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

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

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3

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

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

Step 3.4

Combine and simplify the denominator.

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Step 3.4.1

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

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

Step 3.4.2

Raise $\sqrt{(c + v) (c - v)}$ to the power of $1$ .

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

Step 3.4.3

Raise $\sqrt{(c + v) (c - v)}$ to the power of $1$ .

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

Step 3.4.4

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

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

Step 3.4.5

Add $1$ and $1$ .

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

Step 3.4.6

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

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Step 3.4.6.1

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

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

Step 3.4.6.2

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

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

Step 3.4.6.3

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

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

Step 3.4.6.4

Cancel the common factor of $2$ .

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Step 3.4.6.4.1

Cancel the common factor.

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

Step 3.4.6.4.2

Rewrite the expression.

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

Step 3.4.6.5

Simplify.

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

Step 3.5

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

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