Simplify (-1/2a^3b^4-1/3a^2b^2+2/5a^5b-a^2b^3)÷(-5a^3b^2)

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

Step 1

Rewrite the division as a fraction.

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

Step 2

Simplify the numerator.

Factor $a^{2} b$ out of $- \frac{1}{2} \cdot a^{3} b^{4} - \frac{1}{3} \cdot a^{2} b^{2} + \frac{2}{5} \cdot a^{5} b - a^{2} b^{3}$ .

Factor $a^{2} b$ out of $- \frac{1}{2} \cdot a^{3} b^{4}$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine exponents.

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

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

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

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

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

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

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

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

Rewrite $- 1 b^{2}$ as $- b^{2}$ .

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

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

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

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

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

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

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

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

Multiply $2$ by $3$ .

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

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

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

Multiply $3$ by $2$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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

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

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

Factor $b$ out of $- b \cdot 2$ .

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

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

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

Multiply $3$ by $- 1$ .

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

Multiply $- 1$ by $2$ .

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

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

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

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

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

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

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

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

Multiply $6$ by $5$ .

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

Multiply $\frac{2 a^{3}}{5}$ by $\frac{6}{6}$ .

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

Multiply $5$ by $6$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Rewrite using the commutative property of multiplication.

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

Move $- 2$ to the left of $b$ .

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

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

Move $b^{2}$ .

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

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

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

$\frac{a^{2} b (\frac{(- 3 (b^{2} b^{1}) a - 2 \cdot b) \cdot 5 + 2 a^{3} \cdot 6}{30} - b^{2})}{- 5 a^{3} b^{2}}$

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

$\frac{a^{2} b (\frac{(- 3 b^{2 + 1} a - 2 \cdot b) \cdot 5 + 2 a^{3} \cdot 6}{30} - b^{2})}{- 5 a^{3} b^{2}}$

Add $2$ and $1$ .

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

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

Apply the distributive property.

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

Multiply $5$ by $- 3$ .

$\frac{a^{2} b (\frac{- 15 b^{3} a - 2 b \cdot 5 + 2 a^{3} \cdot 6}{30} - b^{2})}{- 5 a^{3} b^{2}}$

Multiply $5$ by $- 2$ .

$\frac{a^{2} b (\frac{- 15 b^{3} a - 10 b + 2 a^{3} \cdot 6}{30} - b^{2})}{- 5 a^{3} b^{2}}$

Multiply $6$ by $2$ .

$\frac{a^{2} b (\frac{- 15 b^{3} a - 10 b + 12 a^{3}}{30} - b^{2})}{- 5 a^{3} b^{2}}$

To write $- b^{2}$ as a fraction with a common denominator, multiply by $\frac{30}{30}$ .

$\frac{a^{2} b (\frac{- 15 b^{3} a - 10 b + 12 a^{3}}{30} - b^{2} \cdot \frac{30}{30})}{- 5 a^{3} b^{2}}$

Combine $- b^{2}$ and $\frac{30}{30}$ .

$\frac{a^{2} b (\frac{- 15 b^{3} a - 10 b + 12 a^{3}}{30} + \frac{- b^{2} \cdot 30}{30})}{- 5 a^{3} b^{2}}$

Combine the numerators over the common denominator.

$\frac{a^{2} b \frac{- 15 b^{3} a - 10 b + 12 a^{3} - b^{2} \cdot 30}{30}}{- 5 a^{3} b^{2}}$

Multiply $30$ by $- 1$ .

$\frac{a^{2} b \frac{- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2}}{30}}{- 5 a^{3} b^{2}}$

Combine exponents.

Combine $a^{2}$ and $\frac{- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2}}{30}$ .

$\frac{b \frac{a^{2} (- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2})}{30}}{- 5 a^{3} b^{2}}$

Combine $b$ and $\frac{a^{2} (- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2})}{30}$ .

$\frac{\frac{b (a^{2} (- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2}))}{30}}{- 5 a^{3} b^{2}}$

Remove unnecessary parentheses.

$\frac{\frac{b a^{2} (- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2})}{30}}{- 5 a^{3} b^{2}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{b a^{2} (- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2})}{30} \cdot \frac{1}{- 5 a^{3} b^{2}}$

Step 4

Combine.

$\frac{b a^{2} (- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2}) \cdot 1}{30 (- 5 a^{3} b^{2})}$

Step 5

Cancel the common factor of $b$ and $b^{2}$ .

Factor $b$ out of $b a^{2} (- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2}) \cdot 1$ .

$\frac{b ((a^{2} (- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2})) \cdot 1)}{30 (- 5 a^{3} b^{2})}$

Cancel the common factors.

Factor $b$ out of $30 (- 5 a^{3} b^{2})$ .

$\frac{b ((a^{2} (- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2})) \cdot 1)}{b (30 (- 5 a^{3} b))}$

Cancel the common factor.

$\frac{b ((a^{2} (- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2})) \cdot 1)}{b (30 (- 5 a^{3} b))}$

Rewrite the expression.

$\frac{(a^{2} (- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2})) \cdot 1}{30 (- 5 a^{3} b)}$

Step 6

Cancel the common factor of $a^{2}$ and $a^{3}$ .

Factor $a^{2}$ out of $(a^{2} (- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2})) \cdot 1$ .

$\frac{a^{2} ((- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2}) \cdot 1)}{30 (- 5 a^{3} b)}$

Cancel the common factors.

Factor $a^{2}$ out of $30 (- 5 a^{3} b)$ .

$\frac{a^{2} ((- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2}) \cdot 1)}{a^{2} (30 (- 5 a b))}$

Cancel the common factor.

$\frac{a^{2} ((- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2}) \cdot 1)}{a^{2} (30 (- 5 a b))}$

Rewrite the expression.

$\frac{(- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2}) \cdot 1}{30 (- 5 a b)}$

Step 7

Simplify the expression.

Multiply $- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2}$ by $1$ .

$\frac{- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2}}{30 (- 5 a b)}$

Multiply $- 5$ by $30$ .

$\frac{- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2}}{- 150 (a b)}$

Move the negative in front of the fraction.

$- \frac{- 15 b^{3} a - 10 b + 12 a^{3} - 30 b^{2}}{(150) a b}$

Step 8

Factor $- 1$ out of $- 15 b^{3} a$ .

$- \frac{- (15 b^{3} a) - 10 b + 12 a^{3} - 30 b^{2}}{150 a b}$

Step 9

Factor $- 1$ out of $- 10 b$ .

$- \frac{- (15 b^{3} a) - (10 b) + 12 a^{3} - 30 b^{2}}{150 a b}$

Step 10

Factor $- 1$ out of $- (15 b^{3} a) - (10 b)$ .

$- \frac{- (15 b^{3} a + 10 b) + 12 a^{3} - 30 b^{2}}{150 a b}$

Step 11

Factor $- 1$ out of $12 a^{3}$ .

$- \frac{- (15 b^{3} a + 10 b) - (- 12 a^{3}) - 30 b^{2}}{150 a b}$

Step 12

Factor $- 1$ out of $- (15 b^{3} a + 10 b) - (- 12 a^{3})$ .

$- \frac{- (15 b^{3} a + 10 b - 12 a^{3}) - 30 b^{2}}{150 a b}$

Step 13

Factor $- 1$ out of $- 30 b^{2}$ .

$- \frac{- (15 b^{3} a + 10 b - 12 a^{3}) - (30 b^{2})}{150 a b}$

Step 14

Factor $- 1$ out of $- (15 b^{3} a + 10 b - 12 a^{3}) - (30 b^{2})$ .

$- \frac{- (15 b^{3} a + 10 b - 12 a^{3} + 30 b^{2})}{150 a b}$

Step 15

Simplify the expression.

Rewrite $- (15 b^{3} a + 10 b - 12 a^{3} + 30 b^{2})$ as $- 1 (15 b^{3} a + 10 b - 12 a^{3} + 30 b^{2})$ .

$- \frac{- 1 (15 b^{3} a + 10 b - 12 a^{3} + 30 b^{2})}{150 a b}$

Move the negative in front of the fraction.

$- - \frac{15 b^{3} a + 10 b - 12 a^{3} + 30 b^{2}}{150 a b}$

Multiply $- 1$ by $- 1$ .

$1 \frac{15 b^{3} a + 10 b - 12 a^{3} + 30 b^{2}}{150 a b}$

Multiply $\frac{15 b^{3} a + 10 b - 12 a^{3} + 30 b^{2}}{150 a b}$ by $1$ .

$\frac{15 b^{3} a + 10 b - 12 a^{3} + 30 b^{2}}{150 a b}$

Simplify (-1/2a^3b^4-1/3a^2b^2+2/5a^5b-a^2b^3)÷(-5a^3b^2)

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