Simplify a^2b(a^3+5ab-b^3)+2ab(a^4-2a^2b+b^3a)

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

Step 1

Simplify each term.

Apply the distributive property.

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

Simplify.

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

Move $a^{3}$ .

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

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

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

Add $3$ and $2$ .

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

Rewrite using the commutative property of multiplication.

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

Rewrite using the commutative property of multiplication.

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

Simplify each term.

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

Move $a$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $b$ by $b$ by adding the exponents.

Move $b$ .

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

Multiply $b$ by $b$ .

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

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

Move $b^{3}$ .

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

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

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

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

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

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

Add $3$ and $1$ .

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

Apply the distributive property.

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

Simplify.

Multiply $a$ by $a^{4}$ by adding the exponents.

Move $a^{4}$ .

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

Multiply $a^{4}$ by $a$ .

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

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

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

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

Add $4$ and $1$ .

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

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

Move $a^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

Move $a$ .

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

Multiply $a$ by $a$ .

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

Simplify each term.

Multiply $b$ by $b$ by adding the exponents.

Move $b$ .

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

Multiply $b$ by $b$ .

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

Multiply $- 2$ by $2$ .

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

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

Move $b^{3}$ .

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

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

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

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

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

$a^{5} b + 5 a^{3} b^{2} - a^{2} b^{4} + 2 a^{5} b - 4 a^{3} b^{2} + 2 a^{2} b^{3 + 1}$

Add $3$ and $1$ .

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

Step 2

Simplify by adding terms.

Add $a^{5} b$ and $2 a^{5} b$ .

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

Subtract $4 a^{3} b^{2}$ from $5 a^{3} b^{2}$ .

$3 a^{5} b + 1 a^{3} b^{2} - a^{2} b^{4} + 2 a^{2} b^{4}$

Multiply $a^{3}$ by $1$ .

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

Add $- a^{2} b^{4}$ and $2 a^{2} b^{4}$ .

$3 a^{5} b + a^{3} b^{2} + 1 a^{2} b^{4}$

Multiply $a^{2}$ by $1$ .

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

Simplify a^2b(a^3+5ab-b^3)+2ab(a^4-2a^2b+b^3a)

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