Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Expand $(a^{2} + 5 b) (a b - b^{2})$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Apply the distributive property.

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

Step 1.1.3

Apply the distributive property.

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

Step 1.2

Simplify each term.

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Step 1.2.1

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

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Step 1.2.1.1

Move $a$ .

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

Step 1.2.1.2

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

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Step 1.2.1.2.1

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

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

Step 1.2.1.2.2

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

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

Step 1.2.1.3

Add $1$ and $2$ .

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

Step 1.2.2

Rewrite using the commutative property of multiplication.

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

Step 1.2.3

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

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Step 1.2.3.1

Move $b$ .

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

Step 1.2.3.2

Multiply $b$ by $b$ .

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

Step 1.2.4

Rewrite using the commutative property of multiplication.

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

Step 1.2.5

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

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Step 1.2.5.1

Move $b^{2}$ .

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

Step 1.2.5.2

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

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Step 1.2.5.2.1

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

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

Step 1.2.5.2.2

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

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

Step 1.2.5.3

Add $2$ and $1$ .

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

Step 1.2.6

Multiply $5$ by $- 1$ .

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

Step 2

Reorder.

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Step 2.1

Move $b^{2}$ .

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

Step 2.2

Move $5 b^{2}$ .

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