Algebra Examples

$\frac{5 y^{2}}{1 - y^{2}} \div (1 - \frac{1}{1 - y})$

Step 1

Rewrite the division as a fraction.

$\frac{\frac{5 y^{2}}{1 - y^{2}}}{1 - \frac{1}{1 - y}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{5 y^{2}}{1 - y^{2}} \cdot \frac{1}{1 - \frac{1}{1 - y}}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{5 y^{2}}{1^{2} - y^{2}} \cdot \frac{1}{1 - \frac{1}{1 - y}}$

Step 3.2

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 $b = y$ .

$\frac{5 y^{2}}{(1 + y) (1 - y)} \cdot \frac{1}{1 - \frac{1}{1 - y}}$

Step 4

Simplify the denominator.

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Step 4.1

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

$\frac{5 y^{2}}{(1 + y) (1 - y)} \cdot \frac{1}{\frac{1 - y}{1 - y} - \frac{1}{1 - y}}$

Step 4.2

Combine the numerators over the common denominator.

$\frac{5 y^{2}}{(1 + y) (1 - y)} \cdot \frac{1}{\frac{1 - y - 1}{1 - y}}$

Step 4.3

Reorder terms.

$\frac{5 y^{2}}{(1 + y) (1 - y)} \cdot \frac{1}{\frac{- y + 1 - 1}{- y + 1}}$

Step 4.4

Rewrite $\frac{- y + 1 - 1}{- y + 1}$ in a factored form.

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Step 4.4.1

Subtract $1$ from $1$ .

$\frac{5 y^{2}}{(1 + y) (1 - y)} \cdot \frac{1}{\frac{- y + 0}{- y + 1}}$

Step 4.4.2

Add $- y$ and $0$ .

$\frac{5 y^{2}}{(1 + y) (1 - y)} \cdot \frac{1}{\frac{- y}{- y + 1}}$

Step 4.5

Move the negative in front of the fraction.

$\frac{5 y^{2}}{(1 + y) (1 - y)} \cdot \frac{1}{- \frac{y}{- y + 1}}$

Step 5

Simplify terms.

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Step 5.1

Combine.

$\frac{5 y^{2} \cdot 1}{(1 + y) (1 - y) (- \frac{y}{- y + 1})}$

Step 5.2

Cancel the common factor of $1$ and $- 1$ .

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Step 5.2.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{5 y^{2} \cdot (- 1 (- 1))}{(1 + y) (1 - y) (- \frac{y}{- y + 1})}$

Step 5.2.2

Move the negative in front of the fraction.

$- \frac{(5 y^{2}) \cdot - 1}{(1 + y) (1 - y) (- \frac{y}{- y + 1})}$

Step 5.3

Move the negative in front of the fraction.

$- - \frac{5 y^{2}}{(1 + y) (1 - y) (- \frac{y}{- y + 1})}$

Step 6

Simplify the denominator.

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Step 6.1

Remove unnecessary parentheses.

$- - \frac{5 y^{2}}{(1 + y) (1 - y) \cdot - 1 \frac{y}{- y + 1}}$

Step 6.2

Factor out negative.

$- - \frac{5 y^{2}}{- (1 + y) (1 - y) \frac{y}{- y + 1}}$

Step 7

Factor $\frac{y}{- y + 1}$ out of $\frac{5 y^{2}}{- (1 + y) (1 - y) \frac{y}{- y + 1}}$ .

$- - (\frac{- y + 1}{y} \cdot \frac{5 y^{2}}{- (1 + y) (1 - y)})$

Step 8

Cancel the common factor of $y$ .

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Step 8.1

Factor $y$ out of $5 y^{2}$ .

$- - (\frac{- y + 1}{y} \cdot \frac{y (5 y)}{- (1 + y) (1 - y)})$

Step 8.2

Cancel the common factor.

$- - (\frac{- y + 1}{y} \cdot \frac{y (5 y)}{- (1 + y) (1 - y)})$

Step 8.3

Rewrite the expression.

$- - ((- y + 1) \frac{5 y}{- (1 + y) (1 - y)})$

Step 9

Move the negative in front of the fraction.

$- - ((- y + 1) (- \frac{5 y}{(1 + y) (1 - y)}))$

Step 10

Apply the distributive property.

$- - (- y (- \frac{5 y}{(1 + y) (1 - y)}) + 1 (- \frac{5 y}{(1 + y) (1 - y)}))$

Step 11

Multiply $- y (- \frac{5 y}{(1 + y) (1 - y)})$ .

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Step 11.1

Multiply $- 1$ by $- 1$ .

$- - (1 y \frac{5 y}{(1 + y) (1 - y)} + 1 (- \frac{5 y}{(1 + y) (1 - y)}))$

Step 11.2

Multiply $y$ by $1$ .

$- - (y \frac{5 y}{(1 + y) (1 - y)} + 1 (- \frac{5 y}{(1 + y) (1 - y)}))$

Step 11.3

Combine $y$ and $\frac{5 y}{(1 + y) (1 - y)}$ .

$- - (\frac{y (5 y)}{(1 + y) (1 - y)} + 1 (- \frac{5 y}{(1 + y) (1 - y)}))$

Step 11.4

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

$- - (\frac{5 (y^{1} y)}{(1 + y) (1 - y)} + 1 (- \frac{5 y}{(1 + y) (1 - y)}))$

Step 11.5

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

$- - (\frac{5 (y^{1} y^{1})}{(1 + y) (1 - y)} + 1 (- \frac{5 y}{(1 + y) (1 - y)}))$

Step 11.6

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

$- - (\frac{5 y^{1 + 1}}{(1 + y) (1 - y)} + 1 (- \frac{5 y}{(1 + y) (1 - y)}))$

Step 11.7

Add $1$ and $1$ .

$- - (\frac{5 y^{2}}{(1 + y) (1 - y)} + 1 (- \frac{5 y}{(1 + y) (1 - y)}))$

Step 12

Multiply $- \frac{5 y}{(1 + y) (1 - y)}$ by $1$ .

$- - (\frac{5 y^{2}}{(1 + y) (1 - y)} - \frac{5 y}{(1 + y) (1 - y)})$

Step 13

Combine the numerators over the common denominator.

$- - \frac{5 y^{2} - 5 y}{(1 + y) (1 - y)}$

Step 14

Factor $5 y$ out of $5 y^{2} - 5 y$ .

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Step 14.1

Factor $5 y$ out of $5 y^{2}$ .

$- - \frac{5 y (y) - 5 y}{(1 + y) (1 - y)}$

Step 14.2

Factor $5 y$ out of $- 5 y$ .

$- - \frac{5 y (y) + 5 y (- 1)}{(1 + y) (1 - y)}$

Step 14.3

Factor $5 y$ out of $5 y (y) + 5 y (- 1)$ .

$- - \frac{5 y (y - 1)}{(1 + y) (1 - y)}$

Step 15

Cancel the common factor of $y - 1$ and $1 - y$ .

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Step 15.1

Factor $- 1$ out of $y$ .

$- - \frac{5 y (- 1 (- y) - 1)}{(1 + y) (1 - y)}$

Step 15.2

Rewrite $- 1$ as $- 1 (1)$ .

$- - \frac{5 y (- 1 (- y) - 1 (1))}{(1 + y) (1 - y)}$

Step 15.3

Factor $- 1$ out of $- 1 (- y) - 1 (1)$ .

$- - \frac{5 y (- 1 (- y + 1))}{(1 + y) (1 - y)}$

Step 15.4

Reorder terms.

$- - \frac{5 y (- 1 (- y + 1))}{(1 + y) (- y + 1)}$

Step 15.5

Cancel the common factor.

$- - \frac{5 y (- 1 (- y + 1))}{(1 + y) (- y + 1)}$

Step 15.6

Rewrite the expression.

$- - \frac{5 y \cdot (- 1)}{1 + y}$

Step 16

Simplify the expression.

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Step 16.1

Multiply $- 1$ by $5$ .

$- - \frac{- 5 y}{1 + y}$

Step 16.2

Move the negative in front of the fraction.

$- - - \frac{5 y}{1 + y}$

Step 17

Multiply $- - \frac{5 y}{1 + y}$ .

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Step 17.1

Multiply $- 1$ by $- 1$ .

$- (1 \frac{5 y}{1 + y})$

Step 17.2

Multiply $\frac{5 y}{1 + y}$ by $1$ .

$- \frac{5 y}{1 + y}$