Algebra Examples

$(m + 1 - \frac{1}{1 - m}) \div (m - \frac{m^{2}}{m - 1})$

Step 1

Rewrite the division as a fraction.

$\frac{m + 1 - \frac{1}{1 - m}}{m - \frac{m^{2}}{m - 1}}$

Step 2

Multiply the numerator and denominator of the fraction by $(1 - m) (m - 1)$ .

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Step 2.1

Multiply $\frac{m + 1 - \frac{1}{1 - m}}{m - \frac{m^{2}}{m - 1}}$ by $\frac{(1 - m) (m - 1)}{(1 - m) (m - 1)}$ .

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

Step 2.2

Combine.

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

Step 3

Apply the distributive property.

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

Step 4

Simplify by cancelling.

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Step 4.1

Cancel the common factor of $1 - m$ .

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Step 4.1.1

Move the leading negative in $- \frac{1}{1 - m}$ into the numerator.

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $m - 1$ .

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Step 4.2.1

Move the leading negative in $- \frac{m^{2}}{m - 1}$ into the numerator.

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

Step 4.2.2

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

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

Step 4.2.3

Cancel the common factor.

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

Step 4.2.4

Rewrite the expression.

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

Step 5

Simplify the numerator.

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Step 5.1

Factor $m - 1$ out of $(1 - m) (m - 1) m + (1 - m) (m - 1) \cdot 1 + (m - 1) \cdot - 1$ .

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Step 5.1.1

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

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

Step 5.1.2

Factor $m - 1$ out of $(1 - m) (m - 1) \cdot 1$ .

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

Step 5.1.3

Factor $m - 1$ out of $(m - 1) \cdot - 1$ .

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

Step 5.1.4

Factor $m - 1$ out of $(m - 1) ((1 - m) m) + (m - 1) ((1 - m) \cdot 1)$ .

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

Step 5.1.5

Factor $m - 1$ out of $(m - 1) ((1 - m) m + (1 - m) \cdot 1) + (m - 1) (- 1)$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply $m$ by $1$ .

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

Step 5.4

Multiply $m$ by $m$ by adding the exponents.

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Step 5.4.1

Move $m$ .

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

Step 5.4.2

Multiply $m$ by $m$ .

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

Step 5.5

Multiply $1 - m$ by $1$ .

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

Step 5.6

Subtract $m$ from $m$ .

$\frac{(m - 1) (- m^{2} + 1 + 0 - 1)}{(1 - m) (m - 1) m + (1 - m) (- m^{2})}$

Step 5.7

Add $- m^{2}$ and $0$ .

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

Step 5.8

Subtract $1$ from $1$ .

$\frac{(m - 1) (- m^{2} + 0)}{(1 - m) (m - 1) m + (1 - m) (- m^{2})}$

Step 5.9

Add $- m^{2}$ and $0$ .

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

Step 5.10

Factor out negative.

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

Step 6

Simplify the denominator.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

Factor $(1 - m) m$ out of $(1 - m) (- m^{2})$ .

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

Step 6.1.3

Factor $(1 - m) m$ out of $(1 - m) m (m - 1) + (1 - m) m (- m)$ .

$\frac{- (m - 1) m^{2}}{(1 - m) m (m - 1 - m)}$

Step 6.2

Subtract $m$ from $m$ .

$\frac{- (m - 1) m^{2}}{(1 - m) m (0 - 1)}$

Step 6.3

Subtract $1$ from $0$ .

$\frac{- (m - 1) m^{2}}{(1 - m) m \cdot - 1}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Move the negative in front of the fraction.

$- \frac{(m - 1) m^{2}}{(1 - m) m \cdot - 1}$

Step 7.2

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

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Step 7.2.1

Factor $m$ out of $(m - 1) m^{2}$ .

$- \frac{m ((m - 1) m)}{(1 - m) m \cdot - 1}$

Step 7.2.2

Cancel the common factors.

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Step 7.2.2.1

Factor $m$ out of $(1 - m) m \cdot - 1$ .

$- \frac{m ((m - 1) m)}{m ((1 - m) \cdot - 1)}$

Step 7.2.2.2

Cancel the common factor.

$- \frac{m ((m - 1) m)}{m ((1 - m) \cdot - 1)}$

Step 7.2.2.3

Rewrite the expression.

$- \frac{(m - 1) m}{(1 - m) \cdot - 1}$

Step 7.3

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

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Step 7.3.1

Factor $- 1$ out of $m$ .

$- \frac{(- 1 (- m) - 1) m}{(1 - m) \cdot - 1}$

Step 7.3.2

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

$- \frac{(- 1 (- m) - 1 (1)) m}{(1 - m) \cdot - 1}$

Step 7.3.3

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

$- \frac{- 1 (- m + 1) m}{(1 - m) \cdot - 1}$

Step 7.3.4

Reorder terms.

$- \frac{- 1 (- m + 1) m}{(- m + 1) \cdot - 1}$

Step 7.3.5

Cancel the common factor.

$- \frac{- 1 (- m + 1) m}{(- m + 1) \cdot - 1}$

Step 7.3.6

Rewrite the expression.

$- \frac{- 1 m}{- 1}$

Step 7.3.7

Move the negative one from the denominator of $\frac{- 1 m}{- 1}$ .

$- (- 1 \cdot (- 1 m))$

Step 7.4

Rewrite $- 1 m$ as $- m$ .

$- (- 1 \cdot (- m))$

Step 8

Multiply $- 1 (- m)$ .

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Step 8.1

Multiply $- 1$ by $- 1$ .

$- (1 m)$

Step 8.2

Multiply $m$ by $1$ .

$- m$