Algebra Examples

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

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

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

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

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

Step 1.2

Multiply

a

$a$ by

a

$a$ by adding the exponents.

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Step 1.2.1

Move

a

$a$ .

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

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

Step 1.2.2

Multiply

a

$a$ by

a

$a$ .

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

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

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

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

Step 1.3

Multiply

b

$b$ by

b

$b$ by adding the exponents.

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Step 1.3.1

Move

b

$b$ .

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

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

Step 1.3.2

Multiply

b

$b$ by

b

$b$ .

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

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

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

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

Step 1.4

Simplify each term.

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Step 1.4.1

Multiply

4

$4$ by

3

$3$ .

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

$12 a^{2} b + 3 a b^{2} \cdot - 5 + 4 b^{2} (2 a^{2} + 1)$

Step 1.4.2

Multiply

- 5

$- 5$ by

3

$3$ .

12 a^{2} b - 15 a b^{2} + 4 b^{2} (2 a^{2} + 1)

$12 a^{2} b - 15 a b^{2} + 4 b^{2} (2 a^{2} + 1)$

12 a^{2} b - 15 a b^{2} + 4 b^{2} (2 a^{2} + 1)

$12 a^{2} b - 15 a b^{2} + 4 b^{2} (2 a^{2} + 1)$

Step 1.5

Apply the distributive property.

12 a^{2} b - 15 a b^{2} + 4 b^{2} (2 a^{2}) + 4 b^{2} \cdot 1

$12 a^{2} b - 15 a b^{2} + 4 b^{2} (2 a^{2}) + 4 b^{2} \cdot 1$

Step 1.6

Rewrite using the commutative property of multiplication.

12 a^{2} b - 15 a b^{2} + 4 \cdot 2 b^{2} a^{2} + 4 b^{2} \cdot 1

$12 a^{2} b - 15 a b^{2} + 4 \cdot 2 b^{2} a^{2} + 4 b^{2} \cdot 1$

Step 1.7

Multiply

4

$4$ by

1

$1$ .

12 a^{2} b - 15 a b^{2} + 4 \cdot 2 b^{2} a^{2} + 4 b^{2}

$12 a^{2} b - 15 a b^{2} + 4 \cdot 2 b^{2} a^{2} + 4 b^{2}$

Step 1.8

Multiply

4

$4$ by

2

$2$ .

12 a^{2} b - 15 a b^{2} + 8 b^{2} a^{2} + 4 b^{2}

$12 a^{2} b - 15 a b^{2} + 8 b^{2} a^{2} + 4 b^{2}$

12 a^{2} b - 15 a b^{2} + 8 b^{2} a^{2} + 4 b^{2}

$12 a^{2} b - 15 a b^{2} + 8 b^{2} a^{2} + 4 b^{2}$

Step 2

Simplify the expression.

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Step 2.1

Move

b^{2}

$b^{2}$ .

12 a^{2} b - 15 a b^{2} + 8 a^{2} b^{2} + 4 b^{2}

$12 a^{2} b - 15 a b^{2} + 8 a^{2} b^{2} + 4 b^{2}$

Step 2.2

Move

- 15 a b^{2}

$- 15 a b^{2}$ .

12 a^{2} b + 8 a^{2} b^{2} - 15 a b^{2} + 4 b^{2}

$12 a^{2} b + 8 a^{2} b^{2} - 15 a b^{2} + 4 b^{2}$

Step 2.3

Reorder

12 a^{2} b

$12 a^{2} b$ and

8 a^{2} b^{2}

$8 a^{2} b^{2}$ .

8 a^{2} b^{2} + 12 a^{2} b - 15 a b^{2} + 4 b^{2}

$8 a^{2} b^{2} + 12 a^{2} b - 15 a b^{2} + 4 b^{2}$

8 a^{2} b^{2} + 12 a^{2} b - 15 a b^{2} + 4 b^{2}

$8 a^{2} b^{2} + 12 a^{2} b - 15 a b^{2} + 4 b^{2}$