Algebra Examples

$3 a (6 a^{2} - 4 a b^{2}) + 8 a^{2} b^{2} - 2 b^{3}$

Step 1

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$a x^{2} + b x + c$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$3 a (6 a^{2}) + 3 a (- 4 a b^{2}) + 8 a^{2} b^{2} - 2 b^{3}$

Step 2.2

Rewrite using the commutative property of multiplication.

$3 \cdot 6 a \cdot a^{2} + 3 a (- 4 a b^{2}) + 8 a^{2} b^{2} - 2 b^{3}$

Step 2.3

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

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Step 2.3.1

Move $a$ .

$3 \cdot 6 a \cdot a^{2} + 3 (a \cdot a) (- 4 b^{2}) + 8 a^{2} b^{2} - 2 b^{3}$

Step 2.3.2

Multiply $a$ by $a$ .

$3 \cdot 6 a \cdot a^{2} + 3 a^{2} (- 4 b^{2}) + 8 a^{2} b^{2} - 2 b^{3}$

Step 2.4

Simplify each term.

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Step 2.4.1

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

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Step 2.4.1.1

Move $a^{2}$ .

$3 \cdot 6 (a^{2} a) + 3 a^{2} (- 4 b^{2}) + 8 a^{2} b^{2} - 2 b^{3}$

Step 2.4.1.2

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

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Step 2.4.1.2.1

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

$3 \cdot 6 (a^{2} a^{1}) + 3 a^{2} (- 4 b^{2}) + 8 a^{2} b^{2} - 2 b^{3}$

Step 2.4.1.2.2

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

$3 \cdot 6 a^{2 + 1} + 3 a^{2} (- 4 b^{2}) + 8 a^{2} b^{2} - 2 b^{3}$

Step 2.4.1.3

Add $2$ and $1$ .

$3 \cdot 6 a^{3} + 3 a^{2} (- 4 b^{2}) + 8 a^{2} b^{2} - 2 b^{3}$

Step 2.4.2

Multiply $3$ by $6$ .

$18 a^{3} + 3 a^{2} (- 4 b^{2}) + 8 a^{2} b^{2} - 2 b^{3}$

Step 2.4.3

Rewrite using the commutative property of multiplication.

$18 a^{3} + 3 \cdot - 4 a^{2} b^{2} + 8 a^{2} b^{2} - 2 b^{3}$

Step 2.4.4

Multiply $3$ by $- 4$ .

$18 a^{3} - 12 a^{2} b^{2} + 8 a^{2} b^{2} - 2 b^{3}$

Step 3

Simplify by adding terms.

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Step 3.1

Add $- 12 a^{2} b^{2}$ and $8 a^{2} b^{2}$ .

$18 a^{3} - 4 a^{2} b^{2} - 2 b^{3}$

Step 3.2

Reorder $18 a^{3}$ and $- 4 a^{2} b^{2}$ .

$- 4 a^{2} b^{2} + 18 a^{3} - 2 b^{3}$