Algebra Examples

${(2 a^{2} b + 9 a b)}^{2}$

Step 1

Rewrite ${(2 a^{2} b + 9 a b)}^{2}$ as $(2 a^{2} b + 9 a b) (2 a^{2} b + 9 a b)$ .

$(2 a^{2} b + 9 a b) (2 a^{2} b + 9 a b)$

Step 2

Expand $(2 a^{2} b + 9 a b) (2 a^{2} b + 9 a b)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$2 a^{2} b (2 a^{2} b + 9 a b) + 9 a b (2 a^{2} b + 9 a b)$

Step 2.2

Apply the distributive property.

$2 a^{2} b (2 a^{2} b) + 2 a^{2} b (9 a b) + 9 a b (2 a^{2} b + 9 a b)$

Step 2.3

Apply the distributive property.

$2 a^{2} b (2 a^{2} b) + 2 a^{2} b (9 a b) + 9 a b (2 a^{2} b) + 9 a b (9 a b)$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

Move $a^{2}$ .

$2 (a^{2} a^{2}) b (2 b) + 2 a^{2} b (9 a b) + 9 a b (2 a^{2} b) + 9 a b (9 a b)$

Step 3.1.1.2

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

$2 a^{2 + 2} b (2 b) + 2 a^{2} b (9 a b) + 9 a b (2 a^{2} b) + 9 a b (9 a b)$

Step 3.1.1.3

Add $2$ and $2$ .

$2 a^{4} b (2 b) + 2 a^{2} b (9 a b) + 9 a b (2 a^{2} b) + 9 a b (9 a b)$

Step 3.1.2

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

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Step 3.1.2.1

Move $b$ .

$2 a^{4} (b \cdot b) \cdot 2 + 2 a^{2} b (9 a b) + 9 a b (2 a^{2} b) + 9 a b (9 a b)$

Step 3.1.2.2

Multiply $b$ by $b$ .

$2 a^{4} b^{2} \cdot 2 + 2 a^{2} b (9 a b) + 9 a b (2 a^{2} b) + 9 a b (9 a b)$

Step 3.1.3

Multiply $2$ by $2$ .

$4 a^{4} b^{2} + 2 a^{2} b (9 a b) + 9 a b (2 a^{2} b) + 9 a b (9 a b)$

Step 3.1.4

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

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Step 3.1.4.1

Move $a$ .

$4 a^{4} b^{2} + 2 (a \cdot a^{2}) b (9 b) + 9 a b (2 a^{2} b) + 9 a b (9 a b)$

Step 3.1.4.2

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

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Step 3.1.4.2.1

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

$4 a^{4} b^{2} + 2 (a^{1} a^{2}) b (9 b) + 9 a b (2 a^{2} b) + 9 a b (9 a b)$

Step 3.1.4.2.2

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

$4 a^{4} b^{2} + 2 a^{1 + 2} b (9 b) + 9 a b (2 a^{2} b) + 9 a b (9 a b)$

Step 3.1.4.3

Add $1$ and $2$ .

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

Step 3.1.5

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

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Step 3.1.5.1

Move $b$ .

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

Step 3.1.5.2

Multiply $b$ by $b$ .

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

Step 3.1.6

Multiply $9$ by $2$ .

$4 a^{4} b^{2} + 18 a^{3} b^{2} + 9 a b (2 a^{2} b) + 9 a b (9 a b)$

Step 3.1.7

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

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Step 3.1.7.1

Move $a^{2}$ .

$4 a^{4} b^{2} + 18 a^{3} b^{2} + 9 (a^{2} a) b (2 b) + 9 a b (9 a b)$

Step 3.1.7.2

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

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Step 3.1.7.2.1

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

$4 a^{4} b^{2} + 18 a^{3} b^{2} + 9 (a^{2} a^{1}) b (2 b) + 9 a b (9 a b)$

Step 3.1.7.2.2

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

$4 a^{4} b^{2} + 18 a^{3} b^{2} + 9 a^{2 + 1} b (2 b) + 9 a b (9 a b)$

Step 3.1.7.3

Add $2$ and $1$ .

$4 a^{4} b^{2} + 18 a^{3} b^{2} + 9 a^{3} b (2 b) + 9 a b (9 a b)$

Step 3.1.8

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

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Step 3.1.8.1

Move $b$ .

$4 a^{4} b^{2} + 18 a^{3} b^{2} + 9 a^{3} (b \cdot b) \cdot 2 + 9 a b (9 a b)$

Step 3.1.8.2

Multiply $b$ by $b$ .

$4 a^{4} b^{2} + 18 a^{3} b^{2} + 9 a^{3} b^{2} \cdot 2 + 9 a b (9 a b)$

Step 3.1.9

Multiply $2$ by $9$ .

$4 a^{4} b^{2} + 18 a^{3} b^{2} + 18 a^{3} b^{2} + 9 a b (9 a b)$

Step 3.1.10

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

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Step 3.1.10.1

Move $a$ .

$4 a^{4} b^{2} + 18 a^{3} b^{2} + 18 a^{3} b^{2} + 9 (a \cdot a) b (9 b)$

Step 3.1.10.2

Multiply $a$ by $a$ .

$4 a^{4} b^{2} + 18 a^{3} b^{2} + 18 a^{3} b^{2} + 9 a^{2} b (9 b)$

Step 3.1.11

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

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Step 3.1.11.1

Move $b$ .

$4 a^{4} b^{2} + 18 a^{3} b^{2} + 18 a^{3} b^{2} + 9 a^{2} (b \cdot b) \cdot 9$

Step 3.1.11.2

Multiply $b$ by $b$ .

$4 a^{4} b^{2} + 18 a^{3} b^{2} + 18 a^{3} b^{2} + 9 a^{2} b^{2} \cdot 9$

Step 3.1.12

Multiply $9$ by $9$ .

$4 a^{4} b^{2} + 18 a^{3} b^{2} + 18 a^{3} b^{2} + 81 a^{2} b^{2}$

Step 3.2

Add $18 a^{3} b^{2}$ and $18 a^{3} b^{2}$ .

$4 a^{4} b^{2} + 36 a^{3} b^{2} + 81 a^{2} b^{2}$