Pre-Algebra Examples

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

Step 1

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

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

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 $1$ .

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

Step 2.1.1.1.2

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

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

Step 2.1.1.2

Add $1$ and $2$ .

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

Step 2.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.1.3

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

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Step 2.1.3.1

Move $r$ .

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

Step 2.1.3.2

Multiply $r$ by $r$ .

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

Step 2.1.4

Rewrite using the commutative property of multiplication.

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

Step 2.1.5

Multiply $s$ by $s$ by adding the exponents.

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Step 2.1.5.1

Move $s$ .

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

Step 2.1.5.2

Multiply $s$ by $s$ .

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

Step 2.1.6

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

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Step 2.1.6.1

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

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Step 2.1.6.1.1

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

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

Step 2.1.6.1.2

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

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

Step 2.1.6.2

Add $1$ and $2$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

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

$r^{3} + s^{3}$