Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

Combine $a$ and $\frac{1}{3}$ .

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

Step 1.3.2

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.5

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

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

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

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Step 3.1

Move $b^{3}$ .

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

Step 3.2

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

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

Step 4

Multiply $a$ by $1$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

Subtract $5 a$ from $2 a$ .

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

Step 7

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

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

Step 8

Simplify terms.

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Step 8.1

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

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

Step 8.2

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

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

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Step 9.1.1

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

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

Step 9.1.2

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

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

Step 9.1.3

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

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

Step 9.2

Multiply $- 3$ by $3$ .

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

Step 9.3

Apply the distributive property.

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

Step 9.4

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

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Step 9.4.1

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

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Step 9.4.1.1

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

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

Step 9.4.1.2

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

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

Step 9.4.2

Add $1$ and $2$ .

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

Step 9.5

Rewrite using the commutative property of multiplication.

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

Step 10

Simplify with factoring out.

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Step 10.1

Rewrite $- 9$ as $- 1 (9)$ .

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

Step 10.2

Factor $- 1$ out of $b^{3}$ .

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

Step 10.3

Factor $- 1$ out of $- 1 (9) - 1 (- b^{3})$ .

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

Step 10.4

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

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

Step 10.5

Factor $- 1$ out of $- 1 (9 - b^{3}) - (- 2 b a^{2})$ .

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

Step 10.6

Move the negative in front of the fraction.

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