Expand the Trigonometric Expression 2ab^2(3a^2+6a-b^3)

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

Step 1

Apply the distributive property.

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

Step 2

Simplify.

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

Move $a^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

Move $a$ .

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

Multiply $a$ by $a$ .

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

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

Move $b^{3}$ .

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

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

$2 a^{3} b^{2} \cdot 3 + 2 a^{2} b^{2} \cdot 6 + 2 a b^{3 + 2} \cdot - 1$

Add $3$ and $2$ .

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

Step 3

Simplify each term.

Multiply $3$ by $2$ .

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

Multiply $6$ by $2$ .

$6 a^{3} b^{2} + 12 a^{2} b^{2} + 2 a b^{5} \cdot - 1$

Multiply $- 1$ by $2$ .

$6 a^{3} b^{2} + 12 a^{2} b^{2} - 2 a b^{5}$