Pre-Algebra Examples

$(r + g) (r^{2} - r g + g^{2})$

Step 1

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

$r \cdot r^{2} + r (- r g) + r g^{2} + g r^{2} + g (- r g) + g \cdot g^{2}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

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

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Step 2.1.1.1.1

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

$r^{1} r^{2} + r (- r g) + r g^{2} + g r^{2} + g (- r g) + g \cdot g^{2}$

Step 2.1.1.1.2

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

$r^{1 + 2} + r (- r g) + r g^{2} + g r^{2} + g (- r g) + g \cdot g^{2}$

Step 2.1.1.2

Add $1$ and $2$ .

$r^{3} + r (- r g) + r g^{2} + g r^{2} + g (- r g) + g \cdot g^{2}$

Step 2.1.2

Rewrite using the commutative property of multiplication.

$r^{3} - r (r g) + r g^{2} + g r^{2} + g (- r g) + g \cdot g^{2}$

Step 2.1.3

Multiply $r$ by $r$ by adding the exponents.

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Step 2.1.3.1

Move $r$ .

$r^{3} - (r \cdot r) g + r g^{2} + g r^{2} + g (- r g) + g \cdot g^{2}$

Step 2.1.3.2

Multiply $r$ by $r$ .

$r^{3} - r^{2} g + r g^{2} + g r^{2} + g (- r g) + g \cdot g^{2}$

Step 2.1.4

Rewrite using the commutative property of multiplication.

$r^{3} - r^{2} g + r g^{2} + g r^{2} - g (r g) + g \cdot g^{2}$

Step 2.1.5

Multiply $g$ by $g$ by adding the exponents.

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Step 2.1.5.1

Move $g$ .

$r^{3} - r^{2} g + r g^{2} + g r^{2} - (g \cdot g) r + g \cdot g^{2}$

Step 2.1.5.2

Multiply $g$ by $g$ .

$r^{3} - r^{2} g + r g^{2} + g r^{2} - g^{2} r + g \cdot g^{2}$

Step 2.1.6

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

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Step 2.1.6.1

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

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Step 2.1.6.1.1

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

$r^{3} - r^{2} g + r g^{2} + g r^{2} - g^{2} r + g^{1} g^{2}$

Step 2.1.6.1.2

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

$r^{3} - r^{2} g + r g^{2} + g r^{2} - g^{2} r + g^{1 + 2}$

Step 2.1.6.2

Add $1$ and $2$ .

$r^{3} - r^{2} g + r g^{2} + g r^{2} - g^{2} r + g^{3}$

Step 2.2

Combine the opposite terms in $r^{3} - r^{2} g + r g^{2} + g r^{2} - g^{2} r + g^{3}$ .

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Step 2.2.1

Reorder the factors in the terms $- r^{2} g$ and $g r^{2}$ .

$r^{3} - r^{2} g + r g^{2} + r^{2} g - g^{2} r + g^{3}$

Step 2.2.2

Add $- r^{2} g$ and $r^{2} g$ .

$r^{3} + r g^{2} + 0 - g^{2} r + g^{3}$

Step 2.2.3

Add $r^{3} + r g^{2}$ and $0$ .

$r^{3} + r g^{2} - g^{2} r + g^{3}$

Step 2.2.4

Reorder the factors in the terms $r g^{2}$ and $- g^{2} r$ .

$r^{3} + g^{2} r - g^{2} r + g^{3}$

Step 2.2.5

Subtract $g^{2} r$ from $g^{2} r$ .

$r^{3} + 0 + g^{3}$

Step 2.2.6

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

$r^{3} + g^{3}$