Algebra Examples

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

Step 1

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

$a ((a + 9 b) (a + 9 b))$

Step 2

Expand $(a + 9 b) (a + 9 b)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$a (a (a + 9 b) + 9 b (a + 9 b))$

Step 2.2

Apply the distributive property.

$a (a \cdot a + a (9 b) + 9 b (a + 9 b))$

Step 2.3

Apply the distributive property.

$a (a \cdot a + a (9 b) + 9 b a + 9 b (9 b))$

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 $a$ by $a$ .

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

Step 3.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.1.3

Rewrite using the commutative property of multiplication.

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

Step 3.1.4

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

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Step 3.1.4.1

Move $b$ .

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

Step 3.1.4.2

Multiply $b$ by $b$ .

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

Step 3.1.5

Multiply $9$ by $9$ .

$a (a^{2} + 9 a b + 9 b a + 81 b^{2})$

Step 3.2

Add $9 a b$ and $9 b a$ .

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Step 3.2.1

Move $b$ .

$a (a^{2} + 9 a b + 9 a b + 81 b^{2})$

Step 3.2.2

Add $9 a b$ and $9 a b$ .

$a (a^{2} + 18 a b + 81 b^{2})$

Step 4

Apply the distributive property.

$a \cdot a^{2} + a (18 a b) + a (81 b^{2})$

Step 5

Simplify.

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Step 5.1

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

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Step 5.1.1

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

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Step 5.1.1.1

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

$a^{1} a^{2} + a (18 a b) + a (81 b^{2})$

Step 5.1.1.2

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

$a^{1 + 2} + a (18 a b) + a (81 b^{2})$

Step 5.1.2

Add $1$ and $2$ .

$a^{3} + a (18 a b) + a (81 b^{2})$

Step 5.2

Rewrite using the commutative property of multiplication.

$a^{3} + 18 a (a b) + a (81 b^{2})$

Step 5.3

Rewrite using the commutative property of multiplication.

$a^{3} + 18 a (a b) + 81 a b^{2}$

Step 6

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

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Step 6.1

Move $a$ .

$a^{3} + 18 (a \cdot a) b + 81 a b^{2}$

Step 6.2

Multiply $a$ by $a$ .

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