Algebra Examples

$\frac{2}{5} a b^{2} - 16 a + \frac{3}{4} a b^{2} - \frac{6}{5} a^{2} b + 12 a - \frac{3}{5} a^{2} b$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

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

$\frac{a \cdot 2}{5} b^{2} - 16 a + \frac{3}{4} \cdot (a b^{2}) - \frac{6}{5} \cdot (a^{2} b) + 12 a - \frac{3}{5} \cdot (a^{2} b)$

Step 1.1.2

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

$\frac{a \cdot 2 b^{2}}{5} - 16 a + \frac{3}{4} \cdot (a b^{2}) - \frac{6}{5} \cdot (a^{2} b) + 12 a - \frac{3}{5} \cdot (a^{2} b)$

Step 1.2

Move $2$ to the left of $a$ .

$\frac{2 \cdot a b^{2}}{5} - 16 a + \frac{3}{4} \cdot (a b^{2}) - \frac{6}{5} \cdot (a^{2} b) + 12 a - \frac{3}{5} \cdot (a^{2} b)$

Step 1.3

Multiply $\frac{3}{4} (a b^{2})$ .

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Step 1.3.1

Combine $a$ and $\frac{3}{4}$ .

$\frac{2 a b^{2}}{5} - 16 a + \frac{a \cdot 3}{4} b^{2} - \frac{6}{5} \cdot (a^{2} b) + 12 a - \frac{3}{5} \cdot (a^{2} b)$

Step 1.3.2

Combine $\frac{a \cdot 3}{4}$ and $b^{2}$ .

$\frac{2 a b^{2}}{5} - 16 a + \frac{a \cdot 3 b^{2}}{4} - \frac{6}{5} \cdot (a^{2} b) + 12 a - \frac{3}{5} \cdot (a^{2} b)$

Step 1.4

Move $3$ to the left of $a$ .

$\frac{2 a b^{2}}{5} - 16 a + \frac{3 \cdot a b^{2}}{4} - \frac{6}{5} \cdot (a^{2} b) + 12 a - \frac{3}{5} \cdot (a^{2} b)$

Step 1.5

Multiply $- \frac{6}{5} (a^{2} b)$ .

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Step 1.5.1

Combine $a^{2}$ and $\frac{6}{5}$ .

$\frac{2 a b^{2}}{5} - 16 a + \frac{3 a b^{2}}{4} - \frac{a^{2} \cdot 6}{5} b + 12 a - \frac{3}{5} \cdot (a^{2} b)$

Step 1.5.2

Combine $b$ and $\frac{a^{2} \cdot 6}{5}$ .

$\frac{2 a b^{2}}{5} - 16 a + \frac{3 a b^{2}}{4} - \frac{b (a^{2} \cdot 6)}{5} + 12 a - \frac{3}{5} \cdot (a^{2} b)$

Step 1.6

Move $6$ to the left of $b a^{2}$ .

$\frac{2 a b^{2}}{5} - 16 a + \frac{3 a b^{2}}{4} - \frac{6 \cdot (b a^{2})}{5} + 12 a - \frac{3}{5} \cdot (a^{2} b)$

Step 1.7

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

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Step 1.7.1

Combine $a^{2}$ and $\frac{3}{5}$ .

$\frac{2 a b^{2}}{5} - 16 a + \frac{3 a b^{2}}{4} - \frac{6 b a^{2}}{5} + 12 a - \frac{a^{2} \cdot 3}{5} b$

Step 1.7.2

Combine $b$ and $\frac{a^{2} \cdot 3}{5}$ .

$\frac{2 a b^{2}}{5} - 16 a + \frac{3 a b^{2}}{4} - \frac{6 b a^{2}}{5} + 12 a - \frac{b (a^{2} \cdot 3)}{5}$

Step 1.8

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

$\frac{2 a b^{2}}{5} - 16 a + \frac{3 a b^{2}}{4} - \frac{6 b a^{2}}{5} + 12 a - \frac{3 b a^{2}}{5}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$- 16 a + 12 a + \frac{2 a b^{2} - 6 b a^{2} - 3 b a^{2}}{5} + \frac{3 a b^{2}}{4}$

Step 2.2

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

$- 16 a + 12 a + \frac{2 a b^{2} - 9 b a^{2}}{5} + \frac{3 a b^{2}}{4}$

Step 2.3

Factor $a b$ out of $2 a b^{2} - 9 b a^{2}$ .

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Step 2.3.1

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

$- 16 a + 12 a + \frac{a b (2 b) - 9 b a^{2}}{5} + \frac{3 a b^{2}}{4}$

Step 2.3.2

Factor $a b$ out of $- 9 b a^{2}$ .

$- 16 a + 12 a + \frac{a b (2 b) + a b (- 9 a)}{5} + \frac{3 a b^{2}}{4}$

Step 2.3.3

Factor $a b$ out of $a b (2 b) + a b (- 9 a)$ .

$- 16 a + 12 a + \frac{a b (2 b - 9 a)}{5} + \frac{3 a b^{2}}{4}$

Step 2.4

Add $- 16 a$ and $12 a$ .

$- 4 a + \frac{a b (2 b - 9 a)}{5} + \frac{3 a b^{2}}{4}$

Step 3

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

$- 4 a \cdot \frac{5}{5} + \frac{a b (2 b - 9 a)}{5} + \frac{3 a b^{2}}{4}$

Step 4

Simplify terms.

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Step 4.1

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

$\frac{- 4 a \cdot 5}{5} + \frac{a b (2 b - 9 a)}{5} + \frac{3 a b^{2}}{4}$

Step 4.2

Combine the numerators over the common denominator.

$\frac{- 4 a \cdot 5 + a b (2 b - 9 a)}{5} + \frac{3 a b^{2}}{4}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $a$ out of $- 4 a \cdot 5 + a b (2 b - 9 a)$ .

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Step 5.1.1

Factor $a$ out of $- 4 a \cdot 5$ .

$\frac{a (- 4 \cdot 5) + a b (2 b - 9 a)}{5} + \frac{3 a b^{2}}{4}$

Step 5.1.2

Factor $a$ out of $a b (2 b - 9 a)$ .

$\frac{a (- 4 \cdot 5) + a (b (2 b - 9 a))}{5} + \frac{3 a b^{2}}{4}$

Step 5.1.3

Factor $a$ out of $a (- 4 \cdot 5) + a (b (2 b - 9 a))$ .

$\frac{a (- 4 \cdot 5 + b (2 b - 9 a))}{5} + \frac{3 a b^{2}}{4}$

Step 5.2

Multiply $- 4$ by $5$ .

$\frac{a (- 20 + b (2 b - 9 a))}{5} + \frac{3 a b^{2}}{4}$

Step 5.3

Apply the distributive property.

$\frac{a (- 20 + b (2 b) + b (- 9 a))}{5} + \frac{3 a b^{2}}{4}$

Step 5.4

Rewrite using the commutative property of multiplication.

$\frac{a (- 20 + 2 b \cdot b + b (- 9 a))}{5} + \frac{3 a b^{2}}{4}$

Step 5.5

Rewrite using the commutative property of multiplication.

$\frac{a (- 20 + 2 b \cdot b - 9 b a)}{5} + \frac{3 a b^{2}}{4}$

Step 5.6

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

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Step 5.6.1

Move $b$ .

$\frac{a (- 20 + 2 (b \cdot b) - 9 b a)}{5} + \frac{3 a b^{2}}{4}$

Step 5.6.2

Multiply $b$ by $b$ .

$\frac{a (- 20 + 2 b^{2} - 9 b a)}{5} + \frac{3 a b^{2}}{4}$

Step 6

To write $\frac{a (- 20 + 2 b^{2} - 9 b a)}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{a (- 20 + 2 b^{2} - 9 b a)}{5} \cdot \frac{4}{4} + \frac{3 a b^{2}}{4}$

Step 7

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

$\frac{a (- 20 + 2 b^{2} - 9 b a)}{5} \cdot \frac{4}{4} + \frac{3 a b^{2}}{4} \cdot \frac{5}{5}$

Step 8

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1

Multiply $\frac{a (- 20 + 2 b^{2} - 9 b a)}{5}$ by $\frac{4}{4}$ .

$\frac{a (- 20 + 2 b^{2} - 9 b a) \cdot 4}{5 \cdot 4} + \frac{3 a b^{2}}{4} \cdot \frac{5}{5}$

Step 8.2

Multiply $5$ by $4$ .

$\frac{a (- 20 + 2 b^{2} - 9 b a) \cdot 4}{20} + \frac{3 a b^{2}}{4} \cdot \frac{5}{5}$

Step 8.3

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

$\frac{a (- 20 + 2 b^{2} - 9 b a) \cdot 4}{20} + \frac{3 a b^{2} \cdot 5}{4 \cdot 5}$

Step 8.4

Multiply $4$ by $5$ .

$\frac{a (- 20 + 2 b^{2} - 9 b a) \cdot 4}{20} + \frac{3 a b^{2} \cdot 5}{20}$

Step 9

Combine the numerators over the common denominator.

$\frac{a (- 20 + 2 b^{2} - 9 b a) \cdot 4 + 3 a b^{2} \cdot 5}{20}$

Step 10

Simplify the numerator.

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Step 10.1

Factor $a$ out of $a (- 20 + 2 b^{2} - 9 b a) \cdot 4 + 3 a b^{2} \cdot 5$ .

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Step 10.1.1

Factor $a$ out of $a (- 20 + 2 b^{2} - 9 b a) \cdot 4$ .

$\frac{a ((- 20 + 2 b^{2} - 9 b a) \cdot 4) + 3 a b^{2} \cdot 5}{20}$

Step 10.1.2

Factor $a$ out of $3 a b^{2} \cdot 5$ .

$\frac{a ((- 20 + 2 b^{2} - 9 b a) \cdot 4) + a (3 b^{2} \cdot 5)}{20}$

Step 10.1.3

Factor $a$ out of $a ((- 20 + 2 b^{2} - 9 b a) \cdot 4) + a (3 b^{2} \cdot 5)$ .

$\frac{a ((- 20 + 2 b^{2} - 9 b a) \cdot 4 + 3 b^{2} \cdot 5)}{20}$

Step 10.2

Apply the distributive property.

$\frac{a (- 20 \cdot 4 + 2 b^{2} \cdot 4 - 9 b a \cdot 4 + 3 b^{2} \cdot 5)}{20}$

Step 10.3

Simplify.

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Step 10.3.1

Multiply $- 20$ by $4$ .

$\frac{a (- 80 + 2 b^{2} \cdot 4 - 9 b a \cdot 4 + 3 b^{2} \cdot 5)}{20}$

Step 10.3.2

Multiply $4$ by $2$ .

$\frac{a (- 80 + 8 b^{2} - 9 b a \cdot 4 + 3 b^{2} \cdot 5)}{20}$

Step 10.3.3

Multiply $4$ by $- 9$ .

$\frac{a (- 80 + 8 b^{2} - 36 b a + 3 b^{2} \cdot 5)}{20}$

Step 10.4

Multiply $5$ by $3$ .

$\frac{a (- 80 + 8 b^{2} - 36 b a + 15 b^{2})}{20}$

Step 10.5

Add $8 b^{2}$ and $15 b^{2}$ .

$\frac{a (- 80 + 23 b^{2} - 36 b a)}{20}$

Step 11

Simplify with factoring out.

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Step 11.1

Rewrite $- 80$ as $- 1 (80)$ .

$\frac{a (- 1 (80) + 23 b^{2} - 36 b a)}{20}$

Step 11.2

Factor $- 1$ out of $23 b^{2}$ .

$\frac{a (- 1 (80) - (- 23 b^{2}) - 36 b a)}{20}$

Step 11.3

Factor $- 1$ out of $- 1 (80) - (- 23 b^{2})$ .

$\frac{a (- 1 (80 - 23 b^{2}) - 36 b a)}{20}$

Step 11.4

Factor $- 1$ out of $- 36 b a$ .

$\frac{a (- 1 (80 - 23 b^{2}) - (36 b a))}{20}$

Step 11.5

Factor $- 1$ out of $- 1 (80 - 23 b^{2}) - (36 b a)$ .

$\frac{a (- 1 (80 - 23 b^{2} + 36 b a))}{20}$

Step 11.6

Move the negative in front of the fraction.

$- \frac{a (80 - 23 b^{2} + 36 b a)}{20}$