Finite Math Examples

$a - (\frac{a}{3} + \frac{b}{2}) - (\frac{2 (a b)}{5} + 3 a) \cdot (2 a b)$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$a - \frac{a}{3} - \frac{b}{2} - (\frac{2 (a b)}{5} + 3 a) \cdot (2 a b)$

Step 1.2

Apply the distributive property.

$a - \frac{a}{3} - \frac{b}{2} + (- \frac{2 a b}{5} - (3 a)) \cdot (2 a b)$

Step 1.3

Multiply $3$ by $- 1$ .

$a - \frac{a}{3} - \frac{b}{2} + (- \frac{2 a b}{5} - 3 a) \cdot (2 a b)$

Step 1.4

Apply the distributive property.

$a - \frac{a}{3} - \frac{b}{2} - \frac{2 a b}{5} (2 a b) - 3 a (2 a b)$

Step 1.5

Multiply $- \frac{2 a b}{5} (2 a b)$ .

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Step 1.5.1

Multiply $2$ by $- 1$ .

$a - \frac{a}{3} - \frac{b}{2} - 2 \frac{2 a b}{5} (a b) - 3 a (2 a b)$

Step 1.5.2

Combine $- 2$ and $\frac{2 a b}{5}$ .

$a - \frac{a}{3} - \frac{b}{2} + \frac{- 2 (2 a b)}{5} (a b) - 3 a (2 a b)$

Step 1.5.3

Multiply $2$ by $- 2$ .

$a - \frac{a}{3} - \frac{b}{2} + \frac{- 4 (a b)}{5} (a b) - 3 a (2 a b)$

Step 1.5.4

Combine $a$ and $\frac{- 4 (a b)}{5}$ .

$a - \frac{a}{3} - \frac{b}{2} + \frac{a (- 4 (a b))}{5} b - 3 a (2 a b)$

Step 1.5.5

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

$a - \frac{a}{3} - \frac{b}{2} + \frac{- 4 (a^{1} a b)}{5} b - 3 a (2 a b)$

Step 1.5.6

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

$a - \frac{a}{3} - \frac{b}{2} + \frac{- 4 (a^{1} a^{1} b)}{5} b - 3 a (2 a b)$

Step 1.5.7

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

$a - \frac{a}{3} - \frac{b}{2} + \frac{- 4 (a^{1 + 1} b)}{5} b - 3 a (2 a b)$

Step 1.5.8

Add $1$ and $1$ .

$a - \frac{a}{3} - \frac{b}{2} + \frac{- 4 (a^{2} b)}{5} b - 3 a (2 a b)$

Step 1.5.9

Combine $\frac{- 4 (a^{2} b)}{5}$ and $b$ .

$a - \frac{a}{3} - \frac{b}{2} + \frac{- 4 (a^{2} b) b}{5} - 3 a (2 a b)$

Step 1.5.10

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

$a - \frac{a}{3} - \frac{b}{2} + \frac{- 4 (a^{2} (b^{1} b))}{5} - 3 a (2 a b)$

Step 1.5.11

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

$a - \frac{a}{3} - \frac{b}{2} + \frac{- 4 (a^{2} (b^{1} b^{1}))}{5} - 3 a (2 a b)$

Step 1.5.12

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

$a - \frac{a}{3} - \frac{b}{2} + \frac{- 4 (a^{2} b^{1 + 1})}{5} - 3 a (2 a b)$

Step 1.5.13

Add $1$ and $1$ .

$a - \frac{a}{3} - \frac{b}{2} + \frac{- 4 (a^{2} b^{2})}{5} - 3 a (2 a b)$

Step 1.6

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

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Step 1.6.1

Move $a$ .

$a - \frac{a}{3} - \frac{b}{2} + \frac{- 4 (a^{2} b^{2})}{5} - 3 (a \cdot a) (2 b)$

Step 1.6.2

Multiply $a$ by $a$ .

$a - \frac{a}{3} - \frac{b}{2} + \frac{- 4 (a^{2} b^{2})}{5} - 3 a^{2} (2 b)$

Step 1.7

Simplify each term.

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Step 1.7.1

Move the negative in front of the fraction.

$a - \frac{a}{3} - \frac{b}{2} - \frac{4 a^{2} b^{2}}{5} - 3 a^{2} (2 b)$

Step 1.7.2

Rewrite using the commutative property of multiplication.

$a - \frac{a}{3} - \frac{b}{2} - \frac{4 a^{2} b^{2}}{5} - 3 \cdot 2 a^{2} b$

Step 1.7.3

Multiply $- 3$ by $2$ .

$a - \frac{a}{3} - \frac{b}{2} - \frac{4 a^{2} b^{2}}{5} - 6 a^{2} b$

Step 2

To write $a$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a \cdot \frac{3}{3} - \frac{a}{3} - \frac{b}{2} - \frac{4 a^{2} b^{2}}{5} - 6 a^{2} b$

Step 3

Combine $a$ and $\frac{3}{3}$ .

$\frac{a \cdot 3}{3} - \frac{a}{3} - \frac{b}{2} - \frac{4 a^{2} b^{2}}{5} - 6 a^{2} b$

Step 4

Combine the numerators over the common denominator.

$\frac{a \cdot 3 - a}{3} - \frac{b}{2} - \frac{4 a^{2} b^{2}}{5} - 6 a^{2} b$

Step 5

Subtract $a$ from $a \cdot 3$ .

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Step 5.1

Reorder $a$ and $3$ .

$\frac{3 \cdot a - a}{3} - \frac{b}{2} - \frac{4 a^{2} b^{2}}{5} - 6 a^{2} b$

Step 5.2

Subtract $a$ from $3 \cdot a$ .

$\frac{2 \cdot a}{3} - \frac{b}{2} - \frac{4 a^{2} b^{2}}{5} - 6 a^{2} b$

Step 6

To write $- 6 a^{2} b$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2 a}{3} - \frac{4 a^{2} b^{2}}{5} - \frac{b}{2} - 6 a^{2} b \cdot \frac{2}{2}$

Step 7

Simplify terms.

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Step 7.1

Combine $- 6 a^{2} b$ and $\frac{2}{2}$ .

$\frac{2 a}{3} - \frac{4 a^{2} b^{2}}{5} - \frac{b}{2} + \frac{- 6 a^{2} b \cdot 2}{2}$

Step 7.2

Combine the numerators over the common denominator.

$\frac{2 a}{3} - \frac{4 a^{2} b^{2}}{5} + \frac{- b - 6 a^{2} b \cdot 2}{2}$

Step 8

Simplify the numerator.

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Step 8.1

Factor $b$ out of $- b - 6 a^{2} b \cdot 2$ .

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Step 8.1.1

Factor $b$ out of $- b$ .

$\frac{2 a}{3} - \frac{4 a^{2} b^{2}}{5} + \frac{b \cdot - 1 - 6 a^{2} b \cdot 2}{2}$

Step 8.1.2

Factor $b$ out of $- 6 a^{2} b \cdot 2$ .

$\frac{2 a}{3} - \frac{4 a^{2} b^{2}}{5} + \frac{b \cdot - 1 + b (- 6 a^{2} \cdot 2)}{2}$

Step 8.1.3

Factor $b$ out of $b \cdot - 1 + b (- 6 a^{2} \cdot 2)$ .

$\frac{2 a}{3} - \frac{4 a^{2} b^{2}}{5} + \frac{b (- 1 - 6 a^{2} \cdot 2)}{2}$

Step 8.2

Multiply $2$ by $- 6$ .

$\frac{2 a}{3} - \frac{4 a^{2} b^{2}}{5} + \frac{b (- 1 - 12 a^{2})}{2}$

Step 9

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

$- \frac{4 a^{2} b^{2}}{5} + \frac{2 a}{3} \cdot \frac{2}{2} + \frac{b (- 1 - 12 a^{2})}{2}$

Step 10

To write $\frac{b (- 1 - 12 a^{2})}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{4 a^{2} b^{2}}{5} + \frac{2 a}{3} \cdot \frac{2}{2} + \frac{b (- 1 - 12 a^{2})}{2} \cdot \frac{3}{3}$

Step 11

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.1

Multiply $\frac{2 a}{3}$ by $\frac{2}{2}$ .

$- \frac{4 a^{2} b^{2}}{5} + \frac{2 a \cdot 2}{3 \cdot 2} + \frac{b (- 1 - 12 a^{2})}{2} \cdot \frac{3}{3}$

Step 11.2

Multiply $3$ by $2$ .

$- \frac{4 a^{2} b^{2}}{5} + \frac{2 a \cdot 2}{6} + \frac{b (- 1 - 12 a^{2})}{2} \cdot \frac{3}{3}$

Step 11.3

Multiply $\frac{b (- 1 - 12 a^{2})}{2}$ by $\frac{3}{3}$ .

$- \frac{4 a^{2} b^{2}}{5} + \frac{2 a \cdot 2}{6} + \frac{b (- 1 - 12 a^{2}) \cdot 3}{2 \cdot 3}$

Step 11.4

Multiply $2$ by $3$ .

$- \frac{4 a^{2} b^{2}}{5} + \frac{2 a \cdot 2}{6} + \frac{b (- 1 - 12 a^{2}) \cdot 3}{6}$

Step 12

Combine the numerators over the common denominator.

$- \frac{4 a^{2} b^{2}}{5} + \frac{2 a \cdot 2 + b (- 1 - 12 a^{2}) \cdot 3}{6}$

Step 13

Simplify the numerator.

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Step 13.1

Multiply $2$ by $2$ .

$- \frac{4 a^{2} b^{2}}{5} + \frac{4 a + b (- 1 - 12 a^{2}) \cdot 3}{6}$

Step 13.2

Apply the distributive property.

$- \frac{4 a^{2} b^{2}}{5} + \frac{4 a + (b \cdot - 1 + b (- 12 a^{2})) \cdot 3}{6}$

Step 13.3

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

$- \frac{4 a^{2} b^{2}}{5} + \frac{4 a + (- 1 \cdot b + b (- 12 a^{2})) \cdot 3}{6}$

Step 13.4

Rewrite using the commutative property of multiplication.

$- \frac{4 a^{2} b^{2}}{5} + \frac{4 a + (- 1 \cdot b - 12 b a^{2}) \cdot 3}{6}$

Step 13.5

Rewrite $- 1 b$ as $- b$ .

$- \frac{4 a^{2} b^{2}}{5} + \frac{4 a + (- b - 12 b a^{2}) \cdot 3}{6}$

Step 13.6

Apply the distributive property.

$- \frac{4 a^{2} b^{2}}{5} + \frac{4 a - b \cdot 3 - 12 b a^{2} \cdot 3}{6}$

Step 13.7

Multiply $3$ by $- 1$ .

$- \frac{4 a^{2} b^{2}}{5} + \frac{4 a - 3 b - 12 b a^{2} \cdot 3}{6}$

Step 13.8

Multiply $3$ by $- 12$ .

$- \frac{4 a^{2} b^{2}}{5} + \frac{4 a - 3 b - 36 b a^{2}}{6}$

Step 14

To write $- \frac{4 a^{2} b^{2}}{5}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{4 a^{2} b^{2}}{5} \cdot \frac{6}{6} + \frac{4 a - 3 b - 36 b a^{2}}{6}$

Step 15

To write $\frac{4 a - 3 b - 36 b a^{2}}{6}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{4 a^{2} b^{2}}{5} \cdot \frac{6}{6} + \frac{4 a - 3 b - 36 b a^{2}}{6} \cdot \frac{5}{5}$

Step 16

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

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Step 16.1

Multiply $\frac{4 a^{2} b^{2}}{5}$ by $\frac{6}{6}$ .

$- \frac{4 a^{2} b^{2} \cdot 6}{5 \cdot 6} + \frac{4 a - 3 b - 36 b a^{2}}{6} \cdot \frac{5}{5}$

Step 16.2

Multiply $5$ by $6$ .

$- \frac{4 a^{2} b^{2} \cdot 6}{30} + \frac{4 a - 3 b - 36 b a^{2}}{6} \cdot \frac{5}{5}$

Step 16.3

Multiply $\frac{4 a - 3 b - 36 b a^{2}}{6}$ by $\frac{5}{5}$ .

$- \frac{4 a^{2} b^{2} \cdot 6}{30} + \frac{(4 a - 3 b - 36 b a^{2}) \cdot 5}{6 \cdot 5}$

Step 16.4

Multiply $6$ by $5$ .

$- \frac{4 a^{2} b^{2} \cdot 6}{30} + \frac{(4 a - 3 b - 36 b a^{2}) \cdot 5}{30}$

Step 17

Combine the numerators over the common denominator.

$\frac{- 4 a^{2} b^{2} \cdot 6 + (4 a - 3 b - 36 b a^{2}) \cdot 5}{30}$

Step 18

Simplify the numerator.

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Step 18.1

Multiply $6$ by $- 4$ .

$\frac{- 24 a^{2} b^{2} + (4 a - 3 b - 36 b a^{2}) \cdot 5}{30}$

Step 18.2

Apply the distributive property.

$\frac{- 24 a^{2} b^{2} + 4 a \cdot 5 - 3 b \cdot 5 - 36 b a^{2} \cdot 5}{30}$

Step 18.3

Simplify.

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Step 18.3.1

Multiply $5$ by $4$ .

$\frac{- 24 a^{2} b^{2} + 20 a - 3 b \cdot 5 - 36 b a^{2} \cdot 5}{30}$

Step 18.3.2

Multiply $5$ by $- 3$ .

$\frac{- 24 a^{2} b^{2} + 20 a - 15 b - 36 b a^{2} \cdot 5}{30}$

Step 18.3.3

Multiply $5$ by $- 36$ .

$\frac{- 24 a^{2} b^{2} + 20 a - 15 b - 180 b a^{2}}{30}$

Step 19

Simplify with factoring out.

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Step 19.1

Factor $- 1$ out of $- 24 a^{2} b^{2}$ .

$\frac{- (24 a^{2} b^{2}) + 20 a - 15 b - 180 b a^{2}}{30}$

Step 19.2

Factor $- 1$ out of $20 a$ .

$\frac{- (24 a^{2} b^{2}) - (- 20 a) - 15 b - 180 b a^{2}}{30}$

Step 19.3

Factor $- 1$ out of $- (24 a^{2} b^{2}) - (- 20 a)$ .

$\frac{- (24 a^{2} b^{2} - 20 a) - 15 b - 180 b a^{2}}{30}$

Step 19.4

Factor $- 1$ out of $- 15 b$ .

$\frac{- (24 a^{2} b^{2} - 20 a) - (15 b) - 180 b a^{2}}{30}$

Step 19.5

Factor $- 1$ out of $- (24 a^{2} b^{2} - 20 a) - (15 b)$ .

$\frac{- (24 a^{2} b^{2} - 20 a + 15 b) - 180 b a^{2}}{30}$

Step 19.6

Factor $- 1$ out of $- 180 b a^{2}$ .

$\frac{- (24 a^{2} b^{2} - 20 a + 15 b) - (180 b a^{2})}{30}$

Step 19.7

Factor $- 1$ out of $- (24 a^{2} b^{2} - 20 a + 15 b) - (180 b a^{2})$ .

$\frac{- (24 a^{2} b^{2} - 20 a + 15 b + 180 b a^{2})}{30}$

Step 19.8

Simplify the expression.

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Step 19.8.1

Rewrite $- (24 a^{2} b^{2} - 20 a + 15 b + 180 b a^{2})$ as $- 1 (24 a^{2} b^{2} - 20 a + 15 b + 180 b a^{2})$ .

$\frac{- 1 (24 a^{2} b^{2} - 20 a + 15 b + 180 b a^{2})}{30}$

Step 19.8.2

Move the negative in front of the fraction.

$- \frac{24 a^{2} b^{2} - 20 a + 15 b + 180 b a^{2}}{30}$