Algebra Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $a^{2}$ by $a^{4}$ by adding the exponents.

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Step 2.1.1.1

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

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

Step 2.1.1.2

Add $2$ and $4$ .

$a^{6} + a^{2} (2 a^{2}) + a^{2} \cdot 4 - 2 a^{4} - 2 (2 a^{2}) - 2 \cdot 4$

Step 2.1.2

Rewrite using the commutative property of multiplication.

$a^{6} + 2 a^{2} a^{2} + a^{2} \cdot 4 - 2 a^{4} - 2 (2 a^{2}) - 2 \cdot 4$

Step 2.1.3

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

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Step 2.1.3.1

Move $a^{2}$ .

$a^{6} + 2 (a^{2} a^{2}) + a^{2} \cdot 4 - 2 a^{4} - 2 (2 a^{2}) - 2 \cdot 4$

Step 2.1.3.2

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

$a^{6} + 2 a^{2 + 2} + a^{2} \cdot 4 - 2 a^{4} - 2 (2 a^{2}) - 2 \cdot 4$

Step 2.1.3.3

Add $2$ and $2$ .

$a^{6} + 2 a^{4} + a^{2} \cdot 4 - 2 a^{4} - 2 (2 a^{2}) - 2 \cdot 4$

Step 2.1.4

Move $4$ to the left of $a^{2}$ .

$a^{6} + 2 a^{4} + 4 \cdot a^{2} - 2 a^{4} - 2 (2 a^{2}) - 2 \cdot 4$

Step 2.1.5

Multiply $2$ by $- 2$ .

$a^{6} + 2 a^{4} + 4 a^{2} - 2 a^{4} - 4 a^{2} - 2 \cdot 4$

Step 2.1.6

Multiply $- 2$ by $4$ .

$a^{6} + 2 a^{4} + 4 a^{2} - 2 a^{4} - 4 a^{2} - 8$

Step 2.2

Combine the opposite terms in $a^{6} + 2 a^{4} + 4 a^{2} - 2 a^{4} - 4 a^{2} - 8$ .

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Step 2.2.1

Subtract $2 a^{4}$ from $2 a^{4}$ .

$a^{6} + 4 a^{2} + 0 - 4 a^{2} - 8$

Step 2.2.2

Add $a^{6} + 4 a^{2}$ and $0$ .

$a^{6} + 4 a^{2} - 4 a^{2} - 8$

Step 2.2.3

Subtract $4 a^{2}$ from $4 a^{2}$ .

$a^{6} + 0 - 8$

Step 2.2.4

Add $a^{6}$ and $0$ .

$a^{6} - 8$