Simplify (ab^2)/3(6a^3b^3+15a^2b^5-3ab^2-12)

$\frac{a b^{2}}{3} (6 a^{3} b^{3} + 15 a^{2} b^{5} - 3 a b^{2} - 12)$

Apply the distributive property.

$\frac{a b^{2}}{3} (6 a^{3} b^{3}) + \frac{a b^{2}}{3} (15 a^{2} b^{5}) + \frac{a b^{2}}{3} (- 3 a b^{2}) + \frac{a b^{2}}{3} \cdot - 12$

Simplify.

Rewrite using the commutative property of multiplication.

$6 \frac{a b^{2}}{3} (a^{3} b^{3}) + \frac{a b^{2}}{3} (15 a^{2} b^{5}) + \frac{a b^{2}}{3} (- 3 a b^{2}) + \frac{a b^{2}}{3} \cdot - 12$

Rewrite using the commutative property of multiplication.

$6 \frac{a b^{2}}{3} (a^{3} b^{3}) + 15 \frac{a b^{2}}{3} (a^{2} b^{5}) + \frac{a b^{2}}{3} (- 3 a b^{2}) + \frac{a b^{2}}{3} \cdot - 12$

Rewrite using the commutative property of multiplication.

$6 \frac{a b^{2}}{3} (a^{3} b^{3}) + 15 \frac{a b^{2}}{3} (a^{2} b^{5}) - 3 \frac{a b^{2}}{3} (a b^{2}) + \frac{a b^{2}}{3} \cdot - 12$

Cancel the common factor of $3$ .

Factor $3$ out of $- 12$ .

$6 \frac{a b^{2}}{3} (a^{3} b^{3}) + 15 \frac{a b^{2}}{3} (a^{2} b^{5}) - 3 \frac{a b^{2}}{3} (a b^{2}) + \frac{a b^{2}}{3} \cdot (3 (- 4))$

Cancel the common factor.

$6 \frac{a b^{2}}{3} (a^{3} b^{3}) + 15 \frac{a b^{2}}{3} (a^{2} b^{5}) - 3 \frac{a b^{2}}{3} (a b^{2}) + \frac{a b^{2}}{3} \cdot (3 \cdot - 4)$

Rewrite the expression.

$6 \frac{a b^{2}}{3} (a^{3} b^{3}) + 15 \frac{a b^{2}}{3} (a^{2} b^{5}) - 3 \frac{a b^{2}}{3} (a b^{2}) + a b^{2} \cdot - 4$

Simplify each term.

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$3 (2) \frac{a b^{2}}{3} (a^{3} b^{3}) + 15 \frac{a b^{2}}{3} (a^{2} b^{5}) - 3 \frac{a b^{2}}{3} (a b^{2}) + a b^{2} \cdot - 4$

Cancel the common factor.

$3 \cdot 2 \frac{a b^{2}}{3} (a^{3} b^{3}) + 15 \frac{a b^{2}}{3} (a^{2} b^{5}) - 3 \frac{a b^{2}}{3} (a b^{2}) + a b^{2} \cdot - 4$

Rewrite the expression.

$2 (a b^{2}) (a^{3} b^{3}) + 15 \frac{a b^{2}}{3} (a^{2} b^{5}) - 3 \frac{a b^{2}}{3} (a b^{2}) + a b^{2} \cdot - 4$

Multiply $a$ by $a^{3}$ by adding the exponents.

Move $a^{3}$ .

$2 (a^{3} a b^{2}) b^{3} + 15 \frac{a b^{2}}{3} (a^{2} b^{5}) - 3 \frac{a b^{2}}{3} (a b^{2}) + a b^{2} \cdot - 4$

Multiply $a^{3}$ by $a$ .

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

$2 (a^{3} a^{1} b^{2}) b^{3} + 15 \frac{a b^{2}}{3} (a^{2} b^{5}) - 3 \frac{a b^{2}}{3} (a b^{2}) + a b^{2} \cdot - 4$

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

$2 (a^{3 + 1} b^{2}) b^{3} + 15 \frac{a b^{2}}{3} (a^{2} b^{5}) - 3 \frac{a b^{2}}{3} (a b^{2}) + a b^{2} \cdot - 4$

Add $3$ and $1$ .

$2 (a^{4} b^{2}) b^{3} + 15 \frac{a b^{2}}{3} (a^{2} b^{5}) - 3 \frac{a b^{2}}{3} (a b^{2}) + a b^{2} \cdot - 4$

Multiply $b^{2}$ by $b^{3}$ by adding the exponents.

Move $b^{3}$ .

$2 (a^{4} (b^{3} b^{2})) + 15 \frac{a b^{2}}{3} (a^{2} b^{5}) - 3 \frac{a b^{2}}{3} (a b^{2}) + a b^{2} \cdot - 4$

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

$2 (a^{4} b^{3 + 2}) + 15 \frac{a b^{2}}{3} (a^{2} b^{5}) - 3 \frac{a b^{2}}{3} (a b^{2}) + a b^{2} \cdot - 4$

Add $3$ and $2$ .

$2 (a^{4} b^{5}) + 15 \frac{a b^{2}}{3} (a^{2} b^{5}) - 3 \frac{a b^{2}}{3} (a b^{2}) + a b^{2} \cdot - 4$

Cancel the common factor of $3$ .

Factor $3$ out of $15$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Move $a^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Multiply $b^{2}$ by $b^{5}$ by adding the exponents.

Move $b^{5}$ .

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

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

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

Add $5$ and $2$ .

$2 a^{4} b^{5} + 5 (a^{3} b^{7}) - 3 \frac{a b^{2}}{3} (a b^{2}) + a b^{2} \cdot - 4$

Cancel the common factor of $3$ .

Factor $3$ out of $- 3$ .

$2 a^{4} b^{5} + 5 a^{3} b^{7} + 3 (- 1) \frac{a b^{2}}{3} (a b^{2}) + a b^{2} \cdot - 4$

Cancel the common factor.

$2 a^{4} b^{5} + 5 a^{3} b^{7} + 3 \cdot - 1 \frac{a b^{2}}{3} (a b^{2}) + a b^{2} \cdot - 4$

Rewrite the expression.

$2 a^{4} b^{5} + 5 a^{3} b^{7} - (a b^{2}) (a b^{2}) + a b^{2} \cdot - 4$

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

$2 a^{4} b^{5} + 5 a^{3} b^{7} - (a \cdot a b^{2}) b^{2} + a b^{2} \cdot - 4$

Multiply $a$ by $a$ .

$2 a^{4} b^{5} + 5 a^{3} b^{7} - (a^{2} b^{2}) b^{2} + a b^{2} \cdot - 4$

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

Move $b^{2}$ .

$2 a^{4} b^{5} + 5 a^{3} b^{7} - (a^{2} (b^{2} b^{2})) + a b^{2} \cdot - 4$

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

$2 a^{4} b^{5} + 5 a^{3} b^{7} - (a^{2} b^{2 + 2}) + a b^{2} \cdot - 4$

Add $2$ and $2$ .

$2 a^{4} b^{5} + 5 a^{3} b^{7} - (a^{2} b^{4}) + a b^{2} \cdot - 4$

Move $- 4$ to the left of $a b^{2}$ .

$2 a^{4} b^{5} + 5 a^{3} b^{7} - a^{2} b^{4} - 4 a b^{2}$

Simplify (ab^2)/3(6a^3b^3+15a^2b^5-3ab^2-12)

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