Algebra Examples

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

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Cancel the common factor of $1$ and $- 1$ .

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Step 2.1

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

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

Step 2.2

Move the negative in front of the fraction.

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

Step 3

Apply the distributive property.

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

Step 4

Simplify.

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Step 4.1

Rewrite using the commutative property of multiplication.

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

Step 4.2

Cancel the common factor of $a b$ .

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Step 4.2.1

Move the leading negative in $- \frac{1}{a b}$ into the numerator.

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

Step 4.2.2

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

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

Step 4.2.3

Cancel the common factor.

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

Step 4.2.4

Rewrite the expression.

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

Step 4.3

Multiply $- 1$ by $- 1$ .

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

Step 4.4

Multiply $a$ by $1$ .

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

Step 4.5

Cancel the common factor of $b$ .

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Step 4.5.1

Move the leading negative in $- \frac{1}{a b}$ into the numerator.

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

Step 4.5.2

Factor $b$ out of $2 b$ .

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

Step 4.5.3

Factor $b$ out of $a b$ .

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

Step 4.5.4

Cancel the common factor.

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

Step 4.5.5

Rewrite the expression.

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

Step 4.6

Combine $2$ and $\frac{- 1}{a}$ .

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

Step 4.7

Multiply $2$ by $- 1$ .

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

Step 5

Simplify each term.

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Step 5.1

Cancel the common factor of $a b$ .

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Step 5.1.1

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

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

Step 5.1.2

Cancel the common factor.

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

Step 5.1.3

Rewrite the expression.

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

Step 5.2

Move the negative in front of the fraction.

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