Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Combine $a^{3}$ and $\frac{1}{2}$ .

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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Step 4.1.1.1

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

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

Step 4.1.1.2

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

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

Step 4.1.1.3

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

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

Step 4.1.2

Multiply $- 1$ by $2$ .

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

Step 4.1.3

Subtract $2$ from $1$ .

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

Step 4.2

Move $- 1$ to the left of $a^{3} b^{2}$ .

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

Step 4.3

Move the negative in front of the fraction.

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

Step 5

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

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Step 5.1

Move $b^{3}$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Combine the numerators over the common denominator.

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

Step 6

Simplify each term.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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Step 6.1.1.1

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

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

Step 6.1.1.2

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

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

Step 6.1.1.3

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

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

Step 6.1.2

Multiply $- 1$ by $3$ .

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

Step 6.1.3

Subtract $1$ from $- 3$ .

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

Step 6.1.4

Multiply $- 4$ by $2$ .

$- \frac{a^{3} b^{2}}{2} + \frac{- 8 a^{2} b^{3}}{3}$

Step 6.2

Move the negative in front of the fraction.

$- \frac{a^{3} b^{2}}{2} - \frac{8 a^{2} b^{3}}{3}$