Basic Math Examples

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

Step 1

Expand $(3 a + 5 a^{2} b) (3 a - 5 a^{2} b)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.1.2

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

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Step 2.1.2.1

Move $a$ .

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

Step 2.1.2.2

Multiply $a$ by $a$ .

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

Step 2.1.3

Multiply $3$ by $3$ .

$9 a^{2} + 3 a (- 5 a^{2} b) + 5 a^{2} b (3 a) + 5 a^{2} b (- 5 a^{2} b)$

Step 2.1.4

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

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Step 2.1.4.1

Move $a^{2}$ .

$9 a^{2} + 3 (a^{2} a) (- 5 b) + 5 a^{2} b (3 a) + 5 a^{2} b (- 5 a^{2} b)$

Step 2.1.4.2

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

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Step 2.1.4.2.1

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

$9 a^{2} + 3 (a^{2} a^{1}) (- 5 b) + 5 a^{2} b (3 a) + 5 a^{2} b (- 5 a^{2} b)$

Step 2.1.4.2.2

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

$9 a^{2} + 3 a^{2 + 1} (- 5 b) + 5 a^{2} b (3 a) + 5 a^{2} b (- 5 a^{2} b)$

Step 2.1.4.3

Add $2$ and $1$ .

$9 a^{2} + 3 a^{3} (- 5 b) + 5 a^{2} b (3 a) + 5 a^{2} b (- 5 a^{2} b)$

Step 2.1.5

Rewrite using the commutative property of multiplication.

$9 a^{2} + 3 \cdot - 5 a^{3} b + 5 a^{2} b (3 a) + 5 a^{2} b (- 5 a^{2} b)$

Step 2.1.6

Multiply $3$ by $- 5$ .

$9 a^{2} - 15 a^{3} b + 5 a^{2} b (3 a) + 5 a^{2} b (- 5 a^{2} b)$

Step 2.1.7

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

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Step 2.1.7.1

Move $a$ .

$9 a^{2} - 15 a^{3} b + 5 (a \cdot a^{2}) b \cdot 3 + 5 a^{2} b (- 5 a^{2} b)$

Step 2.1.7.2

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

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Step 2.1.7.2.1

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

$9 a^{2} - 15 a^{3} b + 5 (a^{1} a^{2}) b \cdot 3 + 5 a^{2} b (- 5 a^{2} b)$

Step 2.1.7.2.2

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

$9 a^{2} - 15 a^{3} b + 5 a^{1 + 2} b \cdot 3 + 5 a^{2} b (- 5 a^{2} b)$

Step 2.1.7.3

Add $1$ and $2$ .

$9 a^{2} - 15 a^{3} b + 5 a^{3} b \cdot 3 + 5 a^{2} b (- 5 a^{2} b)$

Step 2.1.8

Multiply $3$ by $5$ .

$9 a^{2} - 15 a^{3} b + 15 a^{3} b + 5 a^{2} b (- 5 a^{2} b)$

Step 2.1.9

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

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Step 2.1.9.1

Move $a^{2}$ .

$9 a^{2} - 15 a^{3} b + 15 a^{3} b + 5 (a^{2} a^{2}) b (- 5 b)$

Step 2.1.9.2

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

$9 a^{2} - 15 a^{3} b + 15 a^{3} b + 5 a^{2 + 2} b (- 5 b)$

Step 2.1.9.3

Add $2$ and $2$ .

$9 a^{2} - 15 a^{3} b + 15 a^{3} b + 5 a^{4} b (- 5 b)$

Step 2.1.10

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

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Step 2.1.10.1

Move $b$ .

$9 a^{2} - 15 a^{3} b + 15 a^{3} b + 5 a^{4} (b \cdot b) \cdot - 5$

Step 2.1.10.2

Multiply $b$ by $b$ .

$9 a^{2} - 15 a^{3} b + 15 a^{3} b + 5 a^{4} b^{2} \cdot - 5$

Step 2.1.11

Multiply $- 5$ by $5$ .

$9 a^{2} - 15 a^{3} b + 15 a^{3} b - 25 a^{4} b^{2}$

Step 2.2

Add $- 15 a^{3} b$ and $15 a^{3} b$ .

$9 a^{2} + 0 - 25 a^{4} b^{2}$

Step 2.3

Add $9 a^{2}$ and $0$ .

$9 a^{2} - 25 a^{4} b^{2}$