Basic Math Examples

$a \div (a - b) (a - c) + b \div (b - c) (b - a) + c \div (c - a) (c - b)$

Step 1

Simplify each term.

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Step 1.1

Rewrite the division as a fraction.

$\frac{a}{(a - b) (a - c)} + b \div (b - c) (b - a) + c \div (c - a) (c - b)$

Step 1.2

Rewrite the division as a fraction.

$\frac{a}{(a - b) (a - c)} + \frac{b}{(b - c) (b - a)} + c \div (c - a) (c - b)$

Step 1.3

Rewrite the division as a fraction.

$\frac{a}{(a - b) (a - c)} + \frac{b}{(b - c) (b - a)} + \frac{c}{(c - a) (c - b)}$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $- 1$ out of $b$ .

$\frac{a}{(a - b) (a - c)} + \frac{b}{(b - c) (- 1 (- b) - a)} + \frac{c}{(c - a) (c - b)}$

Step 2.2

Factor $- 1$ out of $- a$ .

$\frac{a}{(a - b) (a - c)} + \frac{b}{(b - c) (- 1 (- b) - (a))} + \frac{c}{(c - a) (c - b)}$

Step 2.3

Factor $- 1$ out of $- 1 (- b) - (a)$ .

$\frac{a}{(a - b) (a - c)} + \frac{b}{(b - c) (- 1 (- b + a))} + \frac{c}{(c - a) (c - b)}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Move a negative from the denominator of $\frac{b}{(b - c) (- 1 (- b + a))}$ to the numerator.

$\frac{a}{(a - b) (a - c)} + \frac{- b}{(b - c) (- b + a)} + \frac{c}{(c - a) (c - b)}$

Step 2.4.2

Reorder terms.

$\frac{a}{(a - b) (a - c)} + \frac{- b}{(b - c) (a - b)} + \frac{c}{(c - a) (c - b)}$

Step 3

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

$\frac{a}{(a - b) (a - c)} \cdot \frac{b - c}{b - c} + \frac{- b}{(b - c) (a - b)} + \frac{c}{(c - a) (c - b)}$

Step 4

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

$\frac{a}{(a - b) (a - c)} \cdot \frac{b - c}{b - c} + \frac{- b}{(b - c) (a - b)} \cdot \frac{a - c}{a - c} + \frac{c}{(c - a) (c - b)}$

Step 5

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

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Step 5.1

Multiply $\frac{a}{(a - b) (a - c)}$ by $\frac{b - c}{b - c}$ .

$\frac{a (b - c)}{(a - b) (a - c) (b - c)} + \frac{- b}{(b - c) (a - b)} \cdot \frac{a - c}{a - c} + \frac{c}{(c - a) (c - b)}$

Step 5.2

Multiply $\frac{- b}{(b - c) (a - b)}$ by $\frac{a - c}{a - c}$ .

$\frac{a (b - c)}{(a - b) (a - c) (b - c)} + \frac{- b (a - c)}{(b - c) (a - b) (a - c)} + \frac{c}{(c - a) (c - b)}$

Step 5.3

Reorder the factors of $(b - c) (a - b) (a - c)$ .

$\frac{a (b - c)}{(a - b) (a - c) (b - c)} + \frac{- b (a - c)}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6

Simplify terms.

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Step 6.1

Combine the numerators over the common denominator.

$\frac{a (b - c) - b (a - c)}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2

Simplify each term.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Apply the distributive property.

$\frac{a b + a (- c) - b (a - c)}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.1.2

Rewrite using the commutative property of multiplication.

$\frac{a b - a c - b (a - c)}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.1.3

Apply the distributive property.

$\frac{a b - a c - b a - b (- c)}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.1.4

Rewrite using the commutative property of multiplication.

$\frac{a b - a c - b a - 1 \cdot - 1 b c}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.1.5

Simplify each term.

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Step 6.2.1.5.1

Multiply $- 1$ by $- 1$ .

$\frac{a b - a c - b a + 1 b c}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.1.5.2

Multiply $b$ by $1$ .

$\frac{a b - a c - b a + b c}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.1.6

Subtract $b a$ from $a b$ .

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Step 6.2.1.6.1

Move $b$ .

$\frac{- a c + a b - a b + b c}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.1.6.2

Subtract $a b$ from $a b$ .

$\frac{- a c + 0 + b c}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.1.7

Add $- a c$ and $0$ .

$\frac{- a c + b c}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.1.8

Factor $c$ out of $- a c + b c$ .

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Step 6.2.1.8.1

Factor $c$ out of $- a c$ .

$\frac{c (- a) + b c}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.1.8.2

Factor $c$ out of $b c$ .

$\frac{c (- a) + c b}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.1.8.3

Factor $c$ out of $c (- a) + c b$ .

$\frac{c (- a + b)}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.2

Cancel the common factor of $- a + b$ and $a - b$ .

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Step 6.2.2.1

Factor $- 1$ out of $- a$ .

$\frac{c (- (a) + b)}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.2.2

Factor $- 1$ out of $b$ .

$\frac{c (- (a) - 1 (- b))}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.2.3

Factor $- 1$ out of $- (a) - 1 (- b)$ .

$\frac{c (- (a - b))}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.2.4

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

$\frac{c (- 1 (a - b))}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.2.5

Cancel the common factor.

$\frac{c (- 1 (a - b))}{(a - b) (a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.2.6

Rewrite the expression.

$\frac{c \cdot (- 1)}{(a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.3

Move $- 1$ to the left of $c$ .

$\frac{- 1 \cdot c}{(a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.2.4

Move the negative in front of the fraction.

$- \frac{c}{(a - c) (b - c)} + \frac{c}{(c - a) (c - b)}$

Step 6.3

Simplify terms.

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Step 6.3.1

Factor $- 1$ out of $c$ .

$- \frac{c}{(a - c) (b - c)} + \frac{c}{(- 1 (- c) - a) (c - b)}$

Step 6.3.2

Factor $- 1$ out of $- a$ .

$- \frac{c}{(a - c) (b - c)} + \frac{c}{(- 1 (- c) - (a)) (c - b)}$

Step 6.3.3

Factor $- 1$ out of $- 1 (- c) - (a)$ .

$- \frac{c}{(a - c) (b - c)} + \frac{c}{- 1 (- c + a) (c - b)}$

Step 6.3.4

Simplify the expression.

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Step 6.3.4.1

Move a negative from the denominator of $\frac{c}{- 1 (- c + a) (c - b)}$ to the numerator.

$- \frac{c}{(a - c) (b - c)} + \frac{- c}{(- c + a) (c - b)}$

Step 6.3.4.2

Reorder terms.

$- \frac{c}{(a - c) (b - c)} + \frac{- c}{(a - c) (c - b)}$

Step 6.3.5

Factor $- 1$ out of $c$ .

$- \frac{c}{(a - c) (b - c)} + \frac{- c}{(a - c) (- 1 (- c) - b)}$

Step 6.3.6

Factor $- 1$ out of $- b$ .

$- \frac{c}{(a - c) (b - c)} + \frac{- c}{(a - c) (- 1 (- c) - (b))}$

Step 6.3.7

Factor $- 1$ out of $- 1 (- c) - (b)$ .

$- \frac{c}{(a - c) (b - c)} + \frac{- c}{(a - c) (- 1 (- c + b))}$

Step 6.3.8

Simplify the expression.

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Step 6.3.8.1

Move a negative from the denominator of $\frac{- c}{(a - c) (- 1 (- c + b))}$ to the numerator.

$- \frac{c}{(a - c) (b - c)} + \frac{- - c}{(a - c) (- c + b)}$

Step 6.3.8.2

Reorder terms.

$- \frac{c}{(a - c) (b - c)} + \frac{- - c}{(a - c) (b - c)}$

Step 6.3.9

Combine the numerators over the common denominator.

$\frac{- c - - c}{(a - c) (b - c)}$

Step 7

Simplify the numerator.

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Step 7.1

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

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Step 7.1.1

Factor $c$ out of $- c$ .

$\frac{c \cdot - 1 - - c}{(a - c) (b - c)}$

Step 7.1.2

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

$\frac{c \cdot - 1 + c (- - 1)}{(a - c) (b - c)}$

Step 7.1.3

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

$\frac{c (- 1 - - 1)}{(a - c) (b - c)}$

Step 7.2

Multiply $- 1$ by $- 1$ .

$\frac{c (- 1 + 1)}{(a - c) (b - c)}$

Step 7.3

Add $- 1$ and $1$ .

$\frac{c \cdot 0}{(a - c) (b - c)}$

Step 8

Multiply by zero.

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Step 8.1

Multiply $c$ by $0$ .

$\frac{0}{(a - c) (b - c)}$

Step 8.2

Divide $0$ by $(a - c) (b - c)$ .

$0$