Basic Math Examples

$\frac{3 a}{a^{2} - 9} - \frac{2 b}{a b + 3 b} - \frac{b}{a b - 3 b}$

Step 1

Factor $b$ out of $a b + 3 b$ .

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Step 1.1

Factor $b$ out of $a b$ .

$\frac{3 a}{a^{2} - 9} - \frac{2 b}{b a + 3 b} - \frac{b}{a b - 3 b}$

Step 1.2

Factor $b$ out of $3 b$ .

$\frac{3 a}{a^{2} - 9} - \frac{2 b}{b a + b \cdot 3} - \frac{b}{a b - 3 b}$

Step 1.3

Factor $b$ out of $b a + b \cdot 3$ .

$\frac{3 a}{a^{2} - 9} - \frac{2 b}{b (a + 3)} - \frac{b}{a b - 3 b}$

Step 2

Cancel the common factor of $b$ .

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Step 2.1

Cancel the common factor.

$\frac{3 a}{a^{2} - 9} - \frac{2 b}{b (a + 3)} - \frac{b}{a b - 3 b}$

Step 2.2

Rewrite the expression.

$\frac{3 a}{a^{2} - 9} - \frac{2}{a + 3} - \frac{b}{a b - 3 b}$

Step 3

Factor $b$ out of $a b - 3 b$ .

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Step 3.1

Factor $b$ out of $a b$ .

$\frac{3 a}{a^{2} - 9} - \frac{2}{a + 3} - \frac{b}{b a - 3 b}$

Step 3.2

Factor $b$ out of $- 3 b$ .

$\frac{3 a}{a^{2} - 9} - \frac{2}{a + 3} - \frac{b}{b a + b \cdot - 3}$

Step 3.3

Factor $b$ out of $b a + b \cdot - 3$ .

$\frac{3 a}{a^{2} - 9} - \frac{2}{a + 3} - \frac{b}{b (a - 3)}$

Step 4

Cancel the common factor of $b$ .

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Step 4.1

Cancel the common factor.

$\frac{3 a}{a^{2} - 9} - \frac{2}{a + 3} - \frac{b}{b (a - 3)}$

Step 4.2

Rewrite the expression.

$\frac{3 a}{a^{2} - 9} - \frac{2}{a + 3} - \frac{1}{a - 3}$

Step 5

To write $\frac{3 a}{a^{2} - 9}$ as a fraction with a common denominator, multiply by $\frac{a + 3}{a + 3}$ .

$\frac{3 a}{a^{2} - 9} \cdot \frac{a + 3}{a + 3} - \frac{2}{a + 3} - \frac{1}{a - 3}$

Step 6

To write $- \frac{2}{a + 3}$ as a fraction with a common denominator, multiply by $\frac{a^{2} - 9}{a^{2} - 9}$ .

$\frac{3 a}{a^{2} - 9} \cdot \frac{a + 3}{a + 3} - \frac{2}{a + 3} \cdot \frac{a^{2} - 9}{a^{2} - 9} - \frac{1}{a - 3}$

Step 7

Write each expression with a common denominator of $(a^{2} - 9) (a + 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

Multiply $\frac{3 a}{a^{2} - 9}$ by $\frac{a + 3}{a + 3}$ .

$\frac{3 a (a + 3)}{(a^{2} - 9) (a + 3)} - \frac{2}{a + 3} \cdot \frac{a^{2} - 9}{a^{2} - 9} - \frac{1}{a - 3}$

Step 7.2

Multiply $\frac{2}{a + 3}$ by $\frac{a^{2} - 9}{a^{2} - 9}$ .

$\frac{3 a (a + 3)}{(a^{2} - 9) (a + 3)} - \frac{2 (a^{2} - 9)}{(a + 3) (a^{2} - 9)} - \frac{1}{a - 3}$

Step 7.3

Reorder the factors of $(a^{2} - 9) (a + 3)$ .

$\frac{3 a (a + 3)}{(a + 3) (a^{2} - 9)} - \frac{2 (a^{2} - 9)}{(a + 3) (a^{2} - 9)} - \frac{1}{a - 3}$

Step 8

Combine the numerators over the common denominator.

$\frac{3 a (a + 3) - 2 (a^{2} - 9)}{(a + 3) (a^{2} - 9)} - \frac{1}{a - 3}$

Step 9

To write $\frac{3 a (a + 3) - 2 (a^{2} - 9)}{(a + 3) (a^{2} - 9)}$ as a fraction with a common denominator, multiply by $\frac{a - 3}{a - 3}$ .

$\frac{3 a (a + 3) - 2 (a^{2} - 9)}{(a + 3) (a^{2} - 9)} \cdot \frac{a - 3}{a - 3} - \frac{1}{a - 3}$

Step 10

To write $- \frac{1}{a - 3}$ as a fraction with a common denominator, multiply by $\frac{(a + 3) (a^{2} - 9)}{(a + 3) (a^{2} - 9)}$ .

$\frac{3 a (a + 3) - 2 (a^{2} - 9)}{(a + 3) (a^{2} - 9)} \cdot \frac{a - 3}{a - 3} - \frac{1}{a - 3} \cdot \frac{(a + 3) (a^{2} - 9)}{(a + 3) (a^{2} - 9)}$

Step 11

Write each expression with a common denominator of $(a + 3) (a^{2} - 9) (a - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.1

Multiply $\frac{3 a (a + 3) - 2 (a^{2} - 9)}{(a + 3) (a^{2} - 9)}$ by $\frac{a - 3}{a - 3}$ .

$\frac{(3 a (a + 3) - 2 (a^{2} - 9)) (a - 3)}{(a + 3) (a^{2} - 9) (a - 3)} - \frac{1}{a - 3} \cdot \frac{(a + 3) (a^{2} - 9)}{(a + 3) (a^{2} - 9)}$

Step 11.2

Multiply $\frac{1}{a - 3}$ by $\frac{(a + 3) (a^{2} - 9)}{(a + 3) (a^{2} - 9)}$ .

$\frac{(3 a (a + 3) - 2 (a^{2} - 9)) (a - 3)}{(a + 3) (a^{2} - 9) (a - 3)} - \frac{(a + 3) (a^{2} - 9)}{(a - 3) ((a + 3) (a^{2} - 9))}$

Step 11.3

Reorder the factors of $(a + 3) (a^{2} - 9) (a - 3)$ .

$\frac{(3 a (a + 3) - 2 (a^{2} - 9)) (a - 3)}{(a + 3) (a - 3) (a^{2} - 9)} - \frac{(a + 3) (a^{2} - 9)}{(a - 3) ((a + 3) (a^{2} - 9))}$

Step 11.4

Reorder the factors of $(a - 3) ((a + 3) (a^{2} - 9))$ .

$\frac{(3 a (a + 3) - 2 (a^{2} - 9)) (a - 3)}{(a + 3) (a - 3) (a^{2} - 9)} - \frac{(a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 12

Combine the numerators over the common denominator.

$\frac{(3 a (a + 3) - 2 (a^{2} - 9)) (a - 3) - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13

Rewrite $\frac{(3 a (a + 3) - 2 (a^{2} - 9)) (a - 3) - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$ in a factored form.

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Step 13.1

Simplify each term.

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Step 13.1.1

Apply the distributive property.

$\frac{(3 a \cdot a + 3 a \cdot 3 - 2 (a^{2} - 9)) (a - 3) - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.1.2

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

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Step 13.1.2.1

Move $a$ .

$\frac{(3 (a \cdot a) + 3 a \cdot 3 - 2 (a^{2} - 9)) (a - 3) - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.1.2.2

Multiply $a$ by $a$ .

$\frac{(3 a^{2} + 3 a \cdot 3 - 2 (a^{2} - 9)) (a - 3) - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.1.3

Multiply $3$ by $3$ .

$\frac{(3 a^{2} + 9 a - 2 (a^{2} - 9)) (a - 3) - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.1.4

Apply the distributive property.

$\frac{(3 a^{2} + 9 a - 2 a^{2} - 2 \cdot - 9) (a - 3) - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.1.5

Multiply $- 2$ by $- 9$ .

$\frac{(3 a^{2} + 9 a - 2 a^{2} + 18) (a - 3) - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.2

Subtract $2 a^{2}$ from $3 a^{2}$ .

$\frac{(a^{2} + 9 a + 18) (a - 3) - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.3

Expand $(a^{2} + 9 a + 18) (a - 3)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{a^{2} a + a^{2} \cdot - 3 + 9 a \cdot a + 9 a \cdot - 3 + 18 a + 18 \cdot - 3 - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.4

Simplify each term.

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Step 13.4.1

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

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Step 13.4.1.1

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

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Step 13.4.1.1.1

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

$\frac{a^{2} a^{1} + a^{2} \cdot - 3 + 9 a \cdot a + 9 a \cdot - 3 + 18 a + 18 \cdot - 3 - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.4.1.1.2

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

$\frac{a^{2 + 1} + a^{2} \cdot - 3 + 9 a \cdot a + 9 a \cdot - 3 + 18 a + 18 \cdot - 3 - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.4.1.2

Add $2$ and $1$ .

$\frac{a^{3} + a^{2} \cdot - 3 + 9 a \cdot a + 9 a \cdot - 3 + 18 a + 18 \cdot - 3 - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.4.2

Move $- 3$ to the left of $a^{2}$ .

$\frac{a^{3} - 3 a^{2} + 9 a \cdot a + 9 a \cdot - 3 + 18 a + 18 \cdot - 3 - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.4.3

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

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Step 13.4.3.1

Move $a$ .

$\frac{a^{3} - 3 a^{2} + 9 (a \cdot a) + 9 a \cdot - 3 + 18 a + 18 \cdot - 3 - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.4.3.2

Multiply $a$ by $a$ .

$\frac{a^{3} - 3 a^{2} + 9 a^{2} + 9 a \cdot - 3 + 18 a + 18 \cdot - 3 - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.4.4

Multiply $- 3$ by $9$ .

$\frac{a^{3} - 3 a^{2} + 9 a^{2} - 27 a + 18 a + 18 \cdot - 3 - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.4.5

Multiply $18$ by $- 3$ .

$\frac{a^{3} - 3 a^{2} + 9 a^{2} - 27 a + 18 a - 54 - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.5

Add $- 3 a^{2}$ and $9 a^{2}$ .

$\frac{a^{3} + 6 a^{2} - 27 a + 18 a - 54 - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.6

Add $- 27 a$ and $18 a$ .

$\frac{a^{3} + 6 a^{2} - 9 a - 54 - (a + 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.7

Apply the distributive property.

$\frac{a^{3} + 6 a^{2} - 9 a - 54 + (- a - 1 \cdot 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.8

Multiply $- 1$ by $3$ .

$\frac{a^{3} + 6 a^{2} - 9 a - 54 + (- a - 3) (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.9

Expand $(- a - 3) (a^{2} - 9)$ using the FOIL Method.

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Step 13.9.1

Apply the distributive property.

$\frac{a^{3} + 6 a^{2} - 9 a - 54 - a (a^{2} - 9) - 3 (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.9.2

Apply the distributive property.

$\frac{a^{3} + 6 a^{2} - 9 a - 54 - a \cdot a^{2} - a \cdot - 9 - 3 (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.9.3

Apply the distributive property.

$\frac{a^{3} + 6 a^{2} - 9 a - 54 - a \cdot a^{2} - a \cdot - 9 - 3 a^{2} - 3 \cdot - 9}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.10

Simplify each term.

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Step 13.10.1

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

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Step 13.10.1.1

Move $a^{2}$ .

$\frac{a^{3} + 6 a^{2} - 9 a - 54 - (a^{2} a) - a \cdot - 9 - 3 a^{2} - 3 \cdot - 9}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.10.1.2

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

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Step 13.10.1.2.1

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

$\frac{a^{3} + 6 a^{2} - 9 a - 54 - (a^{2} a^{1}) - a \cdot - 9 - 3 a^{2} - 3 \cdot - 9}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.10.1.2.2

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

$\frac{a^{3} + 6 a^{2} - 9 a - 54 - a^{2 + 1} - a \cdot - 9 - 3 a^{2} - 3 \cdot - 9}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.10.1.3

Add $2$ and $1$ .

$\frac{a^{3} + 6 a^{2} - 9 a - 54 - a^{3} - a \cdot - 9 - 3 a^{2} - 3 \cdot - 9}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.10.2

Multiply $- 9$ by $- 1$ .

$\frac{a^{3} + 6 a^{2} - 9 a - 54 - a^{3} + 9 a - 3 a^{2} - 3 \cdot - 9}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.10.3

Multiply $- 3$ by $- 9$ .

$\frac{a^{3} + 6 a^{2} - 9 a - 54 - a^{3} + 9 a - 3 a^{2} + 27}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.11

Subtract $a^{3}$ from $a^{3}$ .

$\frac{6 a^{2} - 9 a - 54 + 0 + 9 a - 3 a^{2} + 27}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.12

Add $6 a^{2}$ and $0$ .

$\frac{6 a^{2} - 9 a - 54 + 9 a - 3 a^{2} + 27}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.13

Subtract $3 a^{2}$ from $6 a^{2}$ .

$\frac{3 a^{2} - 9 a - 54 + 9 a + 27}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.14

Add $- 9 a$ and $9 a$ .

$\frac{3 a^{2} + 0 - 54 + 27}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.15

Add $3 a^{2}$ and $0$ .

$\frac{3 a^{2} - 54 + 27}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.16

Add $- 54$ and $27$ .

$\frac{3 a^{2} - 27}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.17

Rewrite $3 a^{2} - 27$ in a factored form.

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Step 13.17.1

Factor $3$ out of $3 a^{2} - 27$ .

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Step 13.17.1.1

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

$\frac{3 (a^{2}) - 27}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.17.1.2

Factor $3$ out of $- 27$ .

$\frac{3 a^{2} + 3 \cdot - 9}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.17.1.3

Factor $3$ out of $3 a^{2} + 3 \cdot - 9$ .

$\frac{3 (a^{2} - 9)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.17.2

Rewrite $9$ as $3^{2}$ .

$\frac{3 (a^{2} - 3^{2})}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.17.3

Factor.

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Step 13.17.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = 3$ .

$\frac{3 ((a + 3) (a - 3))}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.17.3.2

Remove unnecessary parentheses.

$\frac{3 (a + 3) (a - 3)}{(a + 3) (a - 3) (a^{2} - 9)}$

Step 13.18

Rewrite $9$ as $3^{2}$ .

$\frac{3 (a + 3) (a - 3)}{(a + 3) (a - 3) (a^{2} - 3^{2})}$

Step 13.19

Factor.

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Step 13.19.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = 3$ .

$\frac{3 (a + 3) (a - 3)}{(a + 3) (a - 3) ((a + 3) (a - 3))}$

Step 13.19.2

Remove unnecessary parentheses.

$\frac{3 (a + 3) (a - 3)}{(a + 3) (a - 3) (a + 3) (a - 3)}$

Step 13.20

Combine exponents.

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Step 13.20.1

Raise $a + 3$ to the power of $1$ .

$\frac{3 (a + 3) (a - 3)}{{(a + 3)}^{1} (a + 3) (a - 3) (a - 3)}$

Step 13.20.2

Raise $a + 3$ to the power of $1$ .

$\frac{3 (a + 3) (a - 3)}{{(a + 3)}^{1} {(a + 3)}^{1} (a - 3) (a - 3)}$

Step 13.20.3

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

$\frac{3 (a + 3) (a - 3)}{{(a + 3)}^{1 + 1} (a - 3) (a - 3)}$

Step 13.20.4

Add $1$ and $1$ .

$\frac{3 (a + 3) (a - 3)}{{(a + 3)}^{2} (a - 3) (a - 3)}$

Step 13.20.5

Raise $a - 3$ to the power of $1$ .

$\frac{3 (a + 3) (a - 3)}{{(a + 3)}^{2} ({(a - 3)}^{1} (a - 3))}$

Step 13.20.6

Raise $a - 3$ to the power of $1$ .

$\frac{3 (a + 3) (a - 3)}{{(a + 3)}^{2} ({(a - 3)}^{1} {(a - 3)}^{1})}$

Step 13.20.7

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

$\frac{3 (a + 3) (a - 3)}{{(a + 3)}^{2} {(a - 3)}^{1 + 1}}$

Step 13.20.8

Add $1$ and $1$ .

$\frac{3 (a + 3) (a - 3)}{{(a + 3)}^{2} {(a - 3)}^{2}}$

Step 13.21

Reduce the expression $\frac{3 (a + 3) (a - 3)}{{(a + 3)}^{2} {(a - 3)}^{2}}$ by cancelling the common factors.

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Step 13.21.1

Factor $a + 3$ out of $3 (a + 3) (a - 3)$ .

$\frac{(a + 3) (3 (a - 3))}{{(a + 3)}^{2} {(a - 3)}^{2}}$

Step 13.21.2

Factor $a + 3$ out of ${(a + 3)}^{2} {(a - 3)}^{2}$ .

$\frac{(a + 3) (3 (a - 3))}{(a + 3) ((a + 3) {(a - 3)}^{2})}$

Step 13.21.3

Cancel the common factor.

$\frac{(a + 3) (3 (a - 3))}{(a + 3) ((a + 3) {(a - 3)}^{2})}$

Step 13.21.4

Rewrite the expression.

$\frac{3 (a - 3)}{(a + 3) {(a - 3)}^{2}}$

Step 14

Reduce the expression $\frac{3 (a - 3)}{(a + 3) {(a - 3)}^{2}}$ by cancelling the common factors.

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Step 14.1

Factor $a - 3$ out of $3 (a - 3)$ .

$\frac{(a - 3) \cdot 3}{(a + 3) {(a - 3)}^{2}}$

Step 14.2

Factor $a - 3$ out of $(a + 3) {(a - 3)}^{2}$ .

$\frac{(a - 3) \cdot 3}{(a - 3) ((a + 3) (a - 3))}$

Step 14.3

Cancel the common factor.

$\frac{(a - 3) \cdot 3}{(a - 3) ((a + 3) (a - 3))}$

Step 14.4

Rewrite the expression.

$\frac{3}{(a + 3) (a - 3)}$