Basic Math Examples

{(a - 2)}^{2} {(a + 2)}^{2}

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

Step 1

Rewrite

{(a - 2)}^{2}

${(a - 2)}^{2}$ as

(a - 2) (a - 2)

$(a - 2) (a - 2)$ .

(a - 2) (a - 2) {(a + 2)}^{2}

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

Step 2

Expand

(a - 2) (a - 2)

$(a - 2) (a - 2)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

(a (a - 2) - 2 (a - 2)) {(a + 2)}^{2}

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

Step 2.2

Apply the distributive property.

(a \cdot a + a \cdot - 2 - 2 (a - 2)) {(a + 2)}^{2}

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

Step 2.3

Apply the distributive property.

(a \cdot a + a \cdot - 2 - 2 a - 2 \cdot - 2) {(a + 2)}^{2}

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

(a \cdot a + a \cdot - 2 - 2 a - 2 \cdot - 2) {(a + 2)}^{2}

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

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

$a$ by

a

$a$ .

(a^{2} + a \cdot - 2 - 2 a - 2 \cdot - 2) {(a + 2)}^{2}

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

Step 3.1.2

Move

- 2

$- 2$ to the left of

a

$a$ .

(a^{2} - 2 \cdot a - 2 a - 2 \cdot - 2) {(a + 2)}^{2}

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

Step 3.1.3

Multiply

- 2

$- 2$ by

- 2

$- 2$ .

(a^{2} - 2 a - 2 a + 4) {(a + 2)}^{2}

$(a^{2} - 2 a - 2 a + 4) {(a + 2)}^{2}$

(a^{2} - 2 a - 2 a + 4) {(a + 2)}^{2}

$(a^{2} - 2 a - 2 a + 4) {(a + 2)}^{2}$

Step 3.2

Subtract

2 a

$2 a$ from

- 2 a

$- 2 a$ .

(a^{2} - 4 a + 4) {(a + 2)}^{2}

$(a^{2} - 4 a + 4) {(a + 2)}^{2}$

(a^{2} - 4 a + 4) {(a + 2)}^{2}

$(a^{2} - 4 a + 4) {(a + 2)}^{2}$

Step 4

Rewrite

{(a + 2)}^{2}

${(a + 2)}^{2}$ as

(a + 2) (a + 2)

$(a + 2) (a + 2)$ .

(a^{2} - 4 a + 4) ((a + 2) (a + 2))

$(a^{2} - 4 a + 4) ((a + 2) (a + 2))$

Step 5

Expand

(a + 2) (a + 2)

$(a + 2) (a + 2)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

(a^{2} - 4 a + 4) (a (a + 2) + 2 (a + 2))

$(a^{2} - 4 a + 4) (a (a + 2) + 2 (a + 2))$

Step 5.2

Apply the distributive property.

(a^{2} - 4 a + 4) (a \cdot a + a \cdot 2 + 2 (a + 2))

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

Step 5.3

Apply the distributive property.

(a^{2} - 4 a + 4) (a \cdot a + a \cdot 2 + 2 a + 2 \cdot 2)

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

(a^{2} - 4 a + 4) (a \cdot a + a \cdot 2 + 2 a + 2 \cdot 2)

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply

a

$a$ by

a

$a$ .

(a^{2} - 4 a + 4) (a^{2} + a \cdot 2 + 2 a + 2 \cdot 2)

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

Step 6.1.2

Move

2

$2$ to the left of

a

$a$ .

(a^{2} - 4 a + 4) (a^{2} + 2 \cdot a + 2 a + 2 \cdot 2)

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

Step 6.1.3

Multiply

2

$2$ by

2

$2$ .

(a^{2} - 4 a + 4) (a^{2} + 2 a + 2 a + 4)

$(a^{2} - 4 a + 4) (a^{2} + 2 a + 2 a + 4)$

(a^{2} - 4 a + 4) (a^{2} + 2 a + 2 a + 4)

$(a^{2} - 4 a + 4) (a^{2} + 2 a + 2 a + 4)$

Step 6.2

Add

2 a

$2 a$ and

2 a

$2 a$ .

(a^{2} - 4 a + 4) (a^{2} + 4 a + 4)

$(a^{2} - 4 a + 4) (a^{2} + 4 a + 4)$

(a^{2} - 4 a + 4) (a^{2} + 4 a + 4)

$(a^{2} - 4 a + 4) (a^{2} + 4 a + 4)$

Step 7

Expand

(a^{2} - 4 a + 4) (a^{2} + 4 a + 4)

$(a^{2} - 4 a + 4) (a^{2} + 4 a + 4)$ by multiplying each term in the first expression by each term in the second expression.

a^{2} a^{2} + a^{2} (4 a) + a^{2} \cdot 4 - 4 a \cdot a^{2} - 4 a (4 a) - 4 a \cdot 4 + 4 a^{2} + 4 (4 a) + 4 \cdot 4

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

Step 8

Simplify terms.

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Step 8.1

Combine the opposite terms in

a^{2} a^{2} + a^{2} (4 a) + a^{2} \cdot 4 - 4 a \cdot a^{2} - 4 a (4 a) - 4 a \cdot 4 + 4 a^{2} + 4 (4 a) + 4 \cdot 4

$a^{2} a^{2} + a^{2} (4 a) + a^{2} \cdot 4 - 4 a \cdot a^{2} - 4 a (4 a) - 4 a \cdot 4 + 4 a^{2} + 4 (4 a) + 4 \cdot 4$ .

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Step 8.1.1

Reorder the factors in the terms

a^{2} (4 a)

$a^{2} (4 a)$ and

- 4 a \cdot a^{2}

$- 4 a \cdot a^{2}$ .

a^{2} a^{2} + 4 a^{2} a + a^{2} \cdot 4 - 4 a^{2} a - 4 a (4 a) - 4 a \cdot 4 + 4 a^{2} + 4 (4 a) + 4 \cdot 4

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

Step 8.1.2

Subtract

4 a^{2} a

$4 a^{2} a$ from

4 a^{2} a

$4 a^{2} a$ .

a^{2} a^{2} + a^{2} \cdot 4 + 0 - 4 a (4 a) - 4 a \cdot 4 + 4 a^{2} + 4 (4 a) + 4 \cdot 4

$a^{2} a^{2} + a^{2} \cdot 4 + 0 - 4 a (4 a) - 4 a \cdot 4 + 4 a^{2} + 4 (4 a) + 4 \cdot 4$

Step 8.1.3

Add

a^{2} a^{2} + a^{2} \cdot 4

$a^{2} a^{2} + a^{2} \cdot 4$ and

0

$0$ .

a^{2} a^{2} + a^{2} \cdot 4 - 4 a (4 a) - 4 a \cdot 4 + 4 a^{2} + 4 (4 a) + 4 \cdot 4

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

Step 8.1.4

Reorder the factors in the terms

- 4 a \cdot 4

$- 4 a \cdot 4$ and

4 (4 a)

$4 (4 a)$ .

a^{2} a^{2} + a^{2} \cdot 4 - 4 a (4 a) - 4 \cdot 4 a + 4 a^{2} + 4 \cdot 4 a + 4 \cdot 4

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

Step 8.1.5

Add

- 4 \cdot 4 a

$- 4 \cdot 4 a$ and

4 \cdot 4 a

$4 \cdot 4 a$ .

a^{2} a^{2} + a^{2} \cdot 4 - 4 a (4 a) + 4 a^{2} + 0 + 4 \cdot 4

$a^{2} a^{2} + a^{2} \cdot 4 - 4 a (4 a) + 4 a^{2} + 0 + 4 \cdot 4$

Step 8.1.6

Add

a^{2} a^{2} + a^{2} \cdot 4 - 4 a (4 a) + 4 a^{2}

$a^{2} a^{2} + a^{2} \cdot 4 - 4 a (4 a) + 4 a^{2}$ and

0

$0$ .

a^{2} a^{2} + a^{2} \cdot 4 - 4 a (4 a) + 4 a^{2} + 4 \cdot 4

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

Step 8.2

Simplify each term.

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Step 8.2.1

Multiply

$a^{2}$ by

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

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Step 8.2.1.1

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 8.2.1.2

Add

$2$ and

$2$ .

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

Step 8.2.2

Move

$4$ to the left of

$a^{2}$ .

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

Step 8.2.3

Rewrite using the commutative property of multiplication.

$a^{4} + 4 a^{2} - 4 \cdot 4 a \cdot a + 4 a^{2} + 4 \cdot 4$

Step 8.2.4

Multiply

$a$ by

$a$ by adding the exponents.

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Step 8.2.4.1

Move

$a$ .

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

Step 8.2.4.2

Multiply

$a$ by

$a$ .

$a^{4} + 4 a^{2} - 4 \cdot 4 a^{2} + 4 a^{2} + 4 \cdot 4$

Step 8.2.5

Multiply

$- 4$ by

$4$ .

$a^{4} + 4 a^{2} - 16 a^{2} + 4 a^{2} + 4 \cdot 4$

Step 8.2.6

Multiply

$4$ by

$4$ .

$a^{4} + 4 a^{2} - 16 a^{2} + 4 a^{2} + 16$

Step 8.3

Simplify by adding terms.

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Step 8.3.1

Subtract

$16 a^{2}$ from

$4 a^{2}$ .

$a^{4} - 12 a^{2} + 4 a^{2} + 16$

Step 8.3.2

Add

$- 12 a^{2}$ and

$4 a^{2}$ .

$a^{4} - 8 a^{2} + 16$