Basic Math Examples

$\frac{a b (a + b)}{a + c} + \frac{b c (a + b)}{a + c} - \frac{b (a^{2} + a c + b c + b^{2})}{a + c}$

Step 1

Combine the numerators over the common denominator.

$\frac{a b (a + b) + b c (a + b) - b (a^{2} + a c + b c + b^{2})}{a + c}$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$\frac{a b a + a b \cdot b + b c (a + b) - b (a^{2} + a c + b c + b^{2})}{a + c}$

Step 2.2

Multiply $a$ by $a$ by adding the exponents.

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Step 2.2.1

Move $a$ .

$\frac{a \cdot a b + a b \cdot b + b c (a + b) - b (a^{2} + a c + b c + b^{2})}{a + c}$

Step 2.2.2

Multiply $a$ by $a$ .

$\frac{a^{2} b + a b \cdot b + b c (a + b) - b (a^{2} + a c + b c + b^{2})}{a + c}$

Step 2.3

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

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Step 2.3.1

Move $b$ .

$\frac{a^{2} b + a (b \cdot b) + b c (a + b) - b (a^{2} + a c + b c + b^{2})}{a + c}$

Step 2.3.2

Multiply $b$ by $b$ .

$\frac{a^{2} b + a b^{2} + b c (a + b) - b (a^{2} + a c + b c + b^{2})}{a + c}$

Step 2.4

Apply the distributive property.

$\frac{a^{2} b + a b^{2} + b c a + b c b - b (a^{2} + a c + b c + b^{2})}{a + c}$

Step 2.5

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

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Step 2.5.1

Move $b$ .

$\frac{a^{2} b + a b^{2} + b c a + b \cdot b c - b (a^{2} + a c + b c + b^{2})}{a + c}$

Step 2.5.2

Multiply $b$ by $b$ .

$\frac{a^{2} b + a b^{2} + b c a + b^{2} c - b (a^{2} + a c + b c + b^{2})}{a + c}$

Step 2.6

Apply the distributive property.

$\frac{a^{2} b + a b^{2} + b c a + b^{2} c - b a^{2} - b (a c) - b (b c) - b \cdot b^{2}}{a + c}$

Step 2.7

Simplify.

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Step 2.7.1

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

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Step 2.7.1.1

Move $b$ .

$\frac{a^{2} b + a b^{2} + b c a + b^{2} c - b a^{2} - b a c - (b \cdot b) c - b \cdot b^{2}}{a + c}$

Step 2.7.1.2

Multiply $b$ by $b$ .

$\frac{a^{2} b + a b^{2} + b c a + b^{2} c - b a^{2} - b a c - b^{2} c - b \cdot b^{2}}{a + c}$

Step 2.7.2

Multiply $b$ by $b^{2}$ by adding the exponents.

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Step 2.7.2.1

Move $b^{2}$ .

$\frac{a^{2} b + a b^{2} + b c a + b^{2} c - b a^{2} - b a c - b^{2} c - (b^{2} b)}{a + c}$

Step 2.7.2.2

Multiply $b^{2}$ by $b$ .

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Step 2.7.2.2.1

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

$\frac{a^{2} b + a b^{2} + b c a + b^{2} c - b a^{2} - b a c - b^{2} c - (b^{2} b^{1})}{a + c}$

Step 2.7.2.2.2

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

$\frac{a^{2} b + a b^{2} + b c a + b^{2} c - b a^{2} - b a c - b^{2} c - b^{2 + 1}}{a + c}$

Step 2.7.2.3

Add $2$ and $1$ .

$\frac{a^{2} b + a b^{2} + b c a + b^{2} c - b a^{2} - b a c - b^{2} c - b^{3}}{a + c}$

Step 3

Simplify terms.

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Step 3.1

Combine the opposite terms in $a^{2} b + a b^{2} + b c a + b^{2} c - b a^{2} - b a c - b^{2} c - b^{3}$ .

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Step 3.1.1

Reorder the factors in the terms $a^{2} b$ and $- b a^{2}$ .

$\frac{a^{2} b + a b^{2} + b c a + b^{2} c - a^{2} b - b a c - b^{2} c - b^{3}}{a + c}$

Step 3.1.2

Subtract $a^{2} b$ from $a^{2} b$ .

$\frac{a b^{2} + b c a + b^{2} c + 0 - b a c - b^{2} c - b^{3}}{a + c}$

Step 3.1.3

Add $a b^{2} + b c a + b^{2} c$ and $0$ .

$\frac{a b^{2} + b c a + b^{2} c - b a c - b^{2} c - b^{3}}{a + c}$

Step 3.1.4

Reorder the factors in the terms $b c a$ and $- b a c$ .

$\frac{a b^{2} + a b c + b^{2} c - a b c - b^{2} c - b^{3}}{a + c}$

Step 3.1.5

Subtract $a b c$ from $a b c$ .

$\frac{a b^{2} + b^{2} c + 0 - b^{2} c - b^{3}}{a + c}$

Step 3.1.6

Add $a b^{2} + b^{2} c$ and $0$ .

$\frac{a b^{2} + b^{2} c - b^{2} c - b^{3}}{a + c}$

Step 3.1.7

Subtract $b^{2} c$ from $b^{2} c$ .

$\frac{a b^{2} + 0 - b^{3}}{a + c}$

Step 3.1.8

Add $a b^{2}$ and $0$ .

$\frac{a b^{2} - b^{3}}{a + c}$

Step 3.2

Factor $b^{2}$ out of $a b^{2} - b^{3}$ .

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Step 3.2.1

Factor $b^{2}$ out of $a b^{2}$ .

$\frac{b^{2} a - b^{3}}{a + c}$

Step 3.2.2

Factor $b^{2}$ out of $- b^{3}$ .

$\frac{b^{2} a + b^{2} (- b)}{a + c}$

Step 3.2.3

Factor $b^{2}$ out of $b^{2} a + b^{2} (- b)$ .

$\frac{b^{2} (a - b)}{a + c}$