Finite Math Examples

$s \cdot 4 = \frac{- a (10 (1 - r^{4}))}{1 - r}$

Step 1

Rewrite the equation as $\frac{- a \cdot (10 (1 - r^{4}))}{1 - r} = s \cdot 4$ .

$\frac{- a \cdot (10 (1 - r^{4}))}{1 - r} = s \cdot 4$

Step 2

Multiply both sides by $1 - r$ .

$\frac{- a \cdot (10 (1 - r^{4}))}{1 - r} (1 - r) = s \cdot 4 (1 - r)$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{- a \cdot (10 (1 - r^{4}))}{1 - r} (1 - r)$ .

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Step 3.1.1.1

Cancel the common factor of $1 - r$ .

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{- a \cdot (10 (1 - r^{4}))}{1 - r} (1 - r) = s \cdot 4 (1 - r)$

Step 3.1.1.1.2

Rewrite the expression.

$- a \cdot (10 (1 - r^{4})) = s \cdot 4 (1 - r)$

Step 3.1.1.2

Multiply $10$ by $- 1$ .

$- 10 a \cdot (1 - r^{4}) = s \cdot 4 (1 - r)$

Step 3.1.1.3

Apply the distributive property.

$- 10 a \cdot 1 - 10 a (- r^{4}) = s \cdot 4 (1 - r)$

Step 3.1.1.4

Simplify the expression.

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Step 3.1.1.4.1

Multiply $- 10$ by $1$ .

$- 10 a - 10 a (- r^{4}) = s \cdot 4 (1 - r)$

Step 3.1.1.4.2

Rewrite using the commutative property of multiplication.

$- 10 a - 10 \cdot - 1 a r^{4} = s \cdot 4 (1 - r)$

Step 3.1.1.4.3

Multiply $- 10$ by $- 1$ .

$- 10 a + 10 a r^{4} = s \cdot 4 (1 - r)$

Step 3.1.1.4.4

Reorder $- 10 a$ and $10 a r^{4}$ .

$10 a r^{4} - 10 a = s \cdot 4 (1 - r)$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $s \cdot 4 (1 - r)$ .

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Step 3.2.1.1

Move $4$ to the left of $s$ .

$10 a r^{4} - 10 a = 4 \cdot s (1 - r)$

Step 3.2.1.2

Apply the distributive property.

$10 a r^{4} - 10 a = 4 s \cdot 1 + 4 s (- r)$

Step 3.2.1.3

Simplify the expression.

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Step 3.2.1.3.1

Multiply $4$ by $1$ .

$10 a r^{4} - 10 a = 4 s + 4 s (- r)$

Step 3.2.1.3.2

Rewrite using the commutative property of multiplication.

$10 a r^{4} - 10 a = 4 s + 4 \cdot - 1 s r$

Step 3.2.1.3.3

Multiply $4$ by $- 1$ .

$10 a r^{4} - 10 a = 4 s - 4 s r$

Step 3.2.1.3.4

Move $s$ .

$10 a r^{4} - 10 a = 4 s - 4 r s$

Step 3.2.1.3.5

Reorder $4 s$ and $- 4 r s$ .

$10 a r^{4} - 10 a = - 4 r s + 4 s$

Step 4

Solve for $a$ .

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Step 4.1

Factor $10 a$ out of $10 a r^{4} - 10 a$ .

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Step 4.1.1

Factor $10 a$ out of $10 a r^{4}$ .

$10 a (r^{4}) - 10 a = - 4 r s + 4 s$

Step 4.1.2

Factor $10 a$ out of $- 10 a$ .

$10 a (r^{4}) + 10 a (- 1) = - 4 r s + 4 s$

Step 4.1.3

Factor $10 a$ out of $10 a (r^{4}) + 10 a (- 1)$ .

$10 a (r^{4} - 1) = - 4 r s + 4 s$

Step 4.2

Rewrite $r^{4}$ as ${(r^{2})}^{2}$ .

$10 a ({(r^{2})}^{2} - 1) = - 4 r s + 4 s$

Step 4.3

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

$10 a ({(r^{2})}^{2} - 1^{2}) = - 4 r s + 4 s$

Step 4.4

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

$10 a ((r^{2} + 1) (r^{2} - 1)) = - 4 r s + 4 s$

Step 4.5

Factor.

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Step 4.5.1

Simplify.

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Step 4.5.1.1

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

$10 a ((r^{2} + 1) (r^{2} - 1^{2})) = - 4 r s + 4 s$

Step 4.5.1.2

Factor.

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Step 4.5.1.2.1

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

$10 a ((r^{2} + 1) ((r + 1) (r - 1))) = - 4 r s + 4 s$

Step 4.5.1.2.2

Remove unnecessary parentheses.

$10 a ((r^{2} + 1) (r + 1) (r - 1)) = - 4 r s + 4 s$

Step 4.5.2

Remove unnecessary parentheses.

$10 a (r^{2} + 1) (r + 1) (r - 1) = - 4 r s + 4 s$

Step 4.6

Divide each term in $10 a (r^{2} + 1) (r + 1) (r - 1) = - 4 r s + 4 s$ by $10 (r^{2} + 1) (r + 1) (r - 1)$ and simplify.

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Step 4.6.1

Divide each term in $10 a (r^{2} + 1) (r + 1) (r - 1) = - 4 r s + 4 s$ by $10 (r^{2} + 1) (r + 1) (r - 1)$ .

$\frac{10 a (r^{2} + 1) (r + 1) (r - 1)}{10 (r^{2} + 1) (r + 1) (r - 1)} = \frac{- 4 r s}{10 (r^{2} + 1) (r + 1) (r - 1)} + \frac{4 s}{10 (r^{2} + 1) (r + 1) (r - 1)}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $10$ .

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Step 4.6.2.1.1

Cancel the common factor.

$\frac{10 a (r^{2} + 1) (r + 1) (r - 1)}{10 (r^{2} + 1) (r + 1) (r - 1)} = \frac{- 4 r s}{10 (r^{2} + 1) (r + 1) (r - 1)} + \frac{4 s}{10 (r^{2} + 1) (r + 1) (r - 1)}$

Step 4.6.2.1.2

Rewrite the expression.

$\frac{((a (r^{2} + 1)) (r + 1)) (r - 1)}{((r^{2} + 1) (r + 1)) (r - 1)} = \frac{- 4 r s}{10 (r^{2} + 1) (r + 1) (r - 1)} + \frac{4 s}{10 (r^{2} + 1) (r + 1) (r - 1)}$

Step 4.6.2.2

Cancel the common factor of $r^{2} + 1$ .

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Step 4.6.2.2.1

Cancel the common factor.

$\frac{a (r^{2} + 1) (r + 1) (r - 1)}{(r^{2} + 1) (r + 1) (r - 1)} = \frac{- 4 r s}{10 (r^{2} + 1) (r + 1) (r - 1)} + \frac{4 s}{10 (r^{2} + 1) (r + 1) (r - 1)}$

Step 4.6.2.2.2

Rewrite the expression.

$\frac{((a) (r + 1)) (r - 1)}{(r + 1) (r - 1)} = \frac{- 4 r s}{10 (r^{2} + 1) (r + 1) (r - 1)} + \frac{4 s}{10 (r^{2} + 1) (r + 1) (r - 1)}$

Step 4.6.2.3

Cancel the common factor of $r + 1$ .

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Step 4.6.2.3.1

Cancel the common factor.

$\frac{a (r + 1) (r - 1)}{(r + 1) (r - 1)} = \frac{- 4 r s}{10 (r^{2} + 1) (r + 1) (r - 1)} + \frac{4 s}{10 (r^{2} + 1) (r + 1) (r - 1)}$

Step 4.6.2.3.2

Rewrite the expression.

$\frac{(a) (r - 1)}{r - 1} = \frac{- 4 r s}{10 (r^{2} + 1) (r + 1) (r - 1)} + \frac{4 s}{10 (r^{2} + 1) (r + 1) (r - 1)}$

Step 4.6.2.4

Cancel the common factor of $r - 1$ .

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Step 4.6.2.4.1

Cancel the common factor.

$\frac{a (r - 1)}{r - 1} = \frac{- 4 r s}{10 (r^{2} + 1) (r + 1) (r - 1)} + \frac{4 s}{10 (r^{2} + 1) (r + 1) (r - 1)}$

Step 4.6.2.4.2

Divide $a$ by $1$ .

$a = \frac{- 4 r s}{10 (r^{2} + 1) (r + 1) (r - 1)} + \frac{4 s}{10 (r^{2} + 1) (r + 1) (r - 1)}$

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Simplify each term.

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Step 4.6.3.1.1

Cancel the common factor of $- 4$ and $10$ .

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Step 4.6.3.1.1.1

Factor $2$ out of $- 4 r s$ .

$a = \frac{2 (- 2 r s)}{10 (r^{2} + 1) (r + 1) (r - 1)} + \frac{4 s}{10 (r^{2} + 1) (r + 1) (r - 1)}$

Step 4.6.3.1.1.2

Cancel the common factors.

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Step 4.6.3.1.1.2.1

Factor $2$ out of $10 (r^{2} + 1) (r + 1) (r - 1)$ .

$a = \frac{2 (- 2 r s)}{2 (5 (r^{2} + 1) (r + 1) (r - 1))} + \frac{4 s}{10 (r^{2} + 1) (r + 1) (r - 1)}$

Step 4.6.3.1.1.2.2

Cancel the common factor.

$a = \frac{2 (- 2 r s)}{2 (5 (r^{2} + 1) (r + 1) (r - 1))} + \frac{4 s}{10 (r^{2} + 1) (r + 1) (r - 1)}$

Step 4.6.3.1.1.2.3

Rewrite the expression.

$a = \frac{- 2 r s}{5 (r^{2} + 1) (r + 1) (r - 1)} + \frac{4 s}{10 (r^{2} + 1) (r + 1) (r - 1)}$

Step 4.6.3.1.2

Move the negative in front of the fraction.

$a = - \frac{2 r s}{5 (r^{2} + 1) (r + 1) (r - 1)} + \frac{4 s}{10 (r^{2} + 1) (r + 1) (r - 1)}$

Step 4.6.3.1.3

Cancel the common factor of $4$ and $10$ .

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Step 4.6.3.1.3.1

Factor $2$ out of $4 s$ .

$a = - \frac{2 r s}{5 (r^{2} + 1) (r + 1) (r - 1)} + \frac{2 (2 s)}{10 (r^{2} + 1) (r + 1) (r - 1)}$

Step 4.6.3.1.3.2

Cancel the common factors.

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Step 4.6.3.1.3.2.1

Factor $2$ out of $10 (r^{2} + 1) (r + 1) (r - 1)$ .

$a = - \frac{2 r s}{5 (r^{2} + 1) (r + 1) (r - 1)} + \frac{2 (2 s)}{2 (5 (r^{2} + 1) (r + 1) (r - 1))}$

Step 4.6.3.1.3.2.2

Cancel the common factor.

$a = - \frac{2 r s}{5 (r^{2} + 1) (r + 1) (r - 1)} + \frac{2 (2 s)}{2 (5 (r^{2} + 1) (r + 1) (r - 1))}$

Step 4.6.3.1.3.2.3

Rewrite the expression.

$a = - \frac{2 r s}{5 (r^{2} + 1) (r + 1) (r - 1)} + \frac{2 s}{5 (r^{2} + 1) (r + 1) (r - 1)}$