Algebra Examples

$\frac{13 c}{b m - b n} - \frac{12 b}{c n - (c m)}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Factor $b$ out of $b m - b n$ .

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Step 1.1.1.1

Factor $b$ out of $b m$ .

$\frac{13 c}{b (m) - b n} - \frac{12 b}{c n - (c m)}$

Step 1.1.1.2

Factor $b$ out of $- b n$ .

$\frac{13 c}{b (m) + b (- 1 n)} - \frac{12 b}{c n - (c m)}$

Step 1.1.1.3

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

$\frac{13 c}{b (m - 1 n)} - \frac{12 b}{c n - (c m)}$

Step 1.1.2

Rewrite $- 1 n$ as $- n$ .

$\frac{13 c}{b (m - n)} - \frac{12 b}{c n - (c m)}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Factor $c$ out of $c n - c m$ .

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Step 1.2.1.1

Factor $c$ out of $c n$ .

$\frac{13 c}{b (m - n)} - \frac{12 b}{c (n) - c m}$

Step 1.2.1.2

Factor $c$ out of $- c m$ .

$\frac{13 c}{b (m - n)} - \frac{12 b}{c (n) + c (- 1 m)}$

Step 1.2.1.3

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

$\frac{13 c}{b (m - n)} - \frac{12 b}{c (n - 1 m)}$

Step 1.2.2

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

$\frac{13 c}{b (m - n)} - \frac{12 b}{c (n - m)}$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $- 1$ out of $n$ .

$\frac{13 c}{b (m - n)} - \frac{12 b}{c (- 1 (- n) - m)}$

Step 2.2

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

$\frac{13 c}{b (m - n)} - \frac{12 b}{c (- 1 (- n) - (m))}$

Step 2.3

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

$\frac{13 c}{b (m - n)} - \frac{12 b}{c (- 1 (- n + m))}$

Step 2.4

Reorder terms.

$\frac{13 c}{b (m - n)} - \frac{12 b}{c (- 1 (m - n))}$

Step 3

To write $\frac{13 c}{b (m - n)}$ as a fraction with a common denominator, multiply by $\frac{c \cdot - 1}{c \cdot - 1}$ .

$\frac{13 c}{b (m - n)} \cdot \frac{c \cdot - 1}{c \cdot - 1} - \frac{12 b}{c (- 1 (m - n))}$

Step 4

To write $- \frac{12 b}{c (- 1 (m - n))}$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

$\frac{13 c}{b (m - n)} \cdot \frac{c \cdot - 1}{c \cdot - 1} - \frac{12 b}{c (- 1 (m - n))} \cdot \frac{b}{b}$

Step 5

Write each expression with a common denominator of $b (m - n) c \cdot - 1$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

Multiply $\frac{13 c}{b (m - n)}$ by $\frac{c \cdot - 1}{c \cdot - 1}$ .

$\frac{13 c (c \cdot - 1)}{b (m - n) (c \cdot - 1)} - \frac{12 b}{c (- 1 (m - n))} \cdot \frac{b}{b}$

Step 5.2

Multiply $\frac{12 b}{c (- 1 (m - n))}$ by $\frac{b}{b}$ .

$\frac{13 c (c \cdot - 1)}{b (m - n) (c \cdot - 1)} - \frac{12 b \cdot b}{c (- 1 (m - n)) b}$

Step 5.3

Reorder the factors of $b (m - n) (c \cdot - 1)$ .

$\frac{13 c (c \cdot - 1)}{- b c (m - n)} - \frac{12 b \cdot b}{c (- 1 (m - n)) b}$

Step 5.4

Reorder the factors of $c (- 1 (m - n)) b$ .

$\frac{13 c (c \cdot - 1)}{- b c (m - n)} - \frac{12 b \cdot b}{- b c (m - n)}$

Step 6

Combine the numerators over the common denominator.

$\frac{13 c (c \cdot - 1) - 12 b \cdot b}{- b c (m - n)}$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $c$ by $c$ by adding the exponents.

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Step 7.1.1

Move $c$ .

$\frac{13 (c \cdot c) \cdot - 1 - 12 b \cdot b}{- b c (m - n)}$

Step 7.1.2

Multiply $c$ by $c$ .

$\frac{13 c^{2} \cdot - 1 - 12 b \cdot b}{- b c (m - n)}$

Step 7.2

Multiply $- 1$ by $13$ .

$\frac{- 13 c^{2} - 12 b \cdot b}{- b c (m - n)}$

Step 7.3

Multiply $b$ by $b$ by adding the exponents.

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Step 7.3.1

Move $b$ .

$\frac{- 13 c^{2} - 12 (b \cdot b)}{- b c (m - n)}$

Step 7.3.2

Multiply $b$ by $b$ .

$\frac{- 13 c^{2} - 12 b^{2}}{- b c (m - n)}$

Step 8

Simplify with factoring out.

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Step 8.1

Move the negative in front of the fraction.

$- \frac{- 13 c^{2} - 12 b^{2}}{b c (m - n)}$

Step 8.2

Factor $- 1$ out of $- 13 c^{2}$ .

$- \frac{- (13 c^{2}) - 12 b^{2}}{b c (m - n)}$

Step 8.3

Factor $- 1$ out of $- 12 b^{2}$ .

$- \frac{- (13 c^{2}) - (12 b^{2})}{b c (m - n)}$

Step 8.4

Factor $- 1$ out of $- (13 c^{2}) - (12 b^{2})$ .

$- \frac{- (13 c^{2} + 12 b^{2})}{b c (m - n)}$

Step 8.5

Simplify the expression.

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Step 8.5.1

Rewrite $- (13 c^{2} + 12 b^{2})$ as $- 1 (13 c^{2} + 12 b^{2})$ .

$- \frac{- 1 (13 c^{2} + 12 b^{2})}{b c (m - n)}$

Step 8.5.2

Move the negative in front of the fraction.

$- - \frac{13 c^{2} + 12 b^{2}}{b c (m - n)}$

Step 8.5.3

Multiply $- 1$ by $- 1$ .

$1 \frac{13 c^{2} + 12 b^{2}}{b c (m - n)}$

Step 8.5.4

Multiply $\frac{13 c^{2} + 12 b^{2}}{b c (m - n)}$ by $1$ .

$\frac{13 c^{2} + 12 b^{2}}{b c (m - n)}$