Basic Math Examples

$(a + 2) (a^{2} - 2 a + 4) (a^{3} - 8)$

Step 1

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

$(a \cdot a^{2} + a (- 2 a) + a \cdot 4 + 2 a^{2} + 2 (- 2 a) + 2 \cdot 4) (a^{3} - 8)$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

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

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Step 2.1.1.1.1

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

$(a^{1} a^{2} + a (- 2 a) + a \cdot 4 + 2 a^{2} + 2 (- 2 a) + 2 \cdot 4) (a^{3} - 8)$

Step 2.1.1.1.2

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

$(a^{1 + 2} + a (- 2 a) + a \cdot 4 + 2 a^{2} + 2 (- 2 a) + 2 \cdot 4) (a^{3} - 8)$

Step 2.1.1.2

Add $1$ and $2$ .

$(a^{3} + a (- 2 a) + a \cdot 4 + 2 a^{2} + 2 (- 2 a) + 2 \cdot 4) (a^{3} - 8)$

Step 2.1.2

Rewrite using the commutative property of multiplication.

$(a^{3} - 2 a \cdot a + a \cdot 4 + 2 a^{2} + 2 (- 2 a) + 2 \cdot 4) (a^{3} - 8)$

Step 2.1.3

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

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Step 2.1.3.1

Move $a$ .

$(a^{3} - 2 (a \cdot a) + a \cdot 4 + 2 a^{2} + 2 (- 2 a) + 2 \cdot 4) (a^{3} - 8)$

Step 2.1.3.2

Multiply $a$ by $a$ .

$(a^{3} - 2 a^{2} + a \cdot 4 + 2 a^{2} + 2 (- 2 a) + 2 \cdot 4) (a^{3} - 8)$

Step 2.1.4

Move $4$ to the left of $a$ .

$(a^{3} - 2 a^{2} + 4 \cdot a + 2 a^{2} + 2 (- 2 a) + 2 \cdot 4) (a^{3} - 8)$

Step 2.1.5

Multiply $- 2$ by $2$ .

$(a^{3} - 2 a^{2} + 4 a + 2 a^{2} - 4 a + 2 \cdot 4) (a^{3} - 8)$

Step 2.1.6

Multiply $2$ by $4$ .

$(a^{3} - 2 a^{2} + 4 a + 2 a^{2} - 4 a + 8) (a^{3} - 8)$

Step 2.2

Combine the opposite terms in $a^{3} - 2 a^{2} + 4 a + 2 a^{2} - 4 a + 8$ .

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Step 2.2.1

Add $- 2 a^{2}$ and $2 a^{2}$ .

$(a^{3} + 4 a + 0 - 4 a + 8) (a^{3} - 8)$

Step 2.2.2

Add $a^{3} + 4 a$ and $0$ .

$(a^{3} + 4 a - 4 a + 8) (a^{3} - 8)$

Step 2.2.3

Subtract $4 a$ from $4 a$ .

$(a^{3} + 0 + 8) (a^{3} - 8)$

Step 2.2.4

Add $a^{3}$ and $0$ .

$(a^{3} + 8) (a^{3} - 8)$

Step 3

Expand $(a^{3} + 8) (a^{3} - 8)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$a^{3} (a^{3} - 8) + 8 (a^{3} - 8)$

Step 3.2

Apply the distributive property.

$a^{3} a^{3} + a^{3} \cdot - 8 + 8 (a^{3} - 8)$

Step 3.3

Apply the distributive property.

$a^{3} a^{3} + a^{3} \cdot - 8 + 8 a^{3} + 8 \cdot - 8$

Step 4

Simplify terms.

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Step 4.1

Combine the opposite terms in $a^{3} a^{3} + a^{3} \cdot - 8 + 8 a^{3} + 8 \cdot - 8$ .

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Step 4.1.1

Reorder the factors in the terms $a^{3} \cdot - 8$ and $8 a^{3}$ .

$a^{3} a^{3} - 8 a^{3} + 8 a^{3} + 8 \cdot - 8$

Step 4.1.2

Add $- 8 a^{3}$ and $8 a^{3}$ .

$a^{3} a^{3} + 0 + 8 \cdot - 8$

Step 4.1.3

Add $a^{3} a^{3}$ and $0$ .

$a^{3} a^{3} + 8 \cdot - 8$

Step 4.2

Simplify each term.

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Step 4.2.1

Multiply $a^{3}$ by $a^{3}$ by adding the exponents.

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Step 4.2.1.1

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

$a^{3 + 3} + 8 \cdot - 8$

Step 4.2.1.2

Add $3$ and $3$ .

$a^{6} + 8 \cdot - 8$

Step 4.2.2

Multiply $8$ by $- 8$ .

$a^{6} - 64$