Basic Math Examples

${(a^{m} + b^{n})}^{2} (a^{m} - b^{n})$

Step 1

Rewrite ${(a^{m} + b^{n})}^{2}$ as $(a^{m} + b^{n}) (a^{m} + b^{n})$ .

$(a^{m} + b^{n}) (a^{m} + b^{n}) (a^{m} - b^{n})$

Step 2

Expand $(a^{m} + b^{n}) (a^{m} + b^{n})$ using the FOIL Method.

Tap for more steps...

Step 2.1

Apply the distributive property.

$(a^{m} (a^{m} + b^{n}) + b^{n} (a^{m} + b^{n})) (a^{m} - b^{n})$

Step 2.2

Apply the distributive property.

$(a^{m} a^{m} + a^{m} b^{n} + b^{n} (a^{m} + b^{n})) (a^{m} - b^{n})$

Step 2.3

Apply the distributive property.

$(a^{m} a^{m} + a^{m} b^{n} + b^{n} a^{m} + b^{n} b^{n}) (a^{m} - b^{n})$

Step 3

Simplify and combine like terms.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

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

Tap for more steps...

Step 3.1.1.1

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

$(a^{m + m} + a^{m} b^{n} + b^{n} a^{m} + b^{n} b^{n}) (a^{m} - b^{n})$

Step 3.1.1.2

Add $m$ and $m$ .

$(a^{2 m} + a^{m} b^{n} + b^{n} a^{m} + b^{n} b^{n}) (a^{m} - b^{n})$

Step 3.1.2

Multiply $b^{n}$ by $b^{n}$ by adding the exponents.

Tap for more steps...

Step 3.1.2.1

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

$(a^{2 m} + a^{m} b^{n} + b^{n} a^{m} + b^{n + n}) (a^{m} - b^{n})$

Step 3.1.2.2

Add $n$ and $n$ .

$(a^{2 m} + a^{m} b^{n} + b^{n} a^{m} + b^{2 n}) (a^{m} - b^{n})$

Step 3.2

Add $a^{m} b^{n}$ and $b^{n} a^{m}$ .

Tap for more steps...

Step 3.2.1

Reorder $b^{n}$ and $a^{m}$ .

$(a^{2 m} + a^{m} b^{n} + a^{m} b^{n} + b^{2 n}) (a^{m} - b^{n})$

Step 3.2.2

Add $a^{m} b^{n}$ and $a^{m} b^{n}$ .

$(a^{2 m} + 2 a^{m} b^{n} + b^{2 n}) (a^{m} - b^{n})$

Step 4

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

$a^{2 m} a^{m} + a^{2 m} (- b^{n}) + 2 a^{m} b^{n} a^{m} + 2 a^{m} b^{n} (- b^{n}) + b^{2 n} a^{m} + b^{2 n} (- b^{n})$

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Simplify each term.

Tap for more steps...

Step 5.1.1

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

Tap for more steps...

Step 5.1.1.1

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

$a^{2 m + m} + a^{2 m} (- b^{n}) + 2 a^{m} b^{n} a^{m} + 2 a^{m} b^{n} (- b^{n}) + b^{2 n} a^{m} + b^{2 n} (- b^{n})$

Step 5.1.1.2

Add $2 m$ and $m$ .

$a^{3 m} + a^{2 m} (- b^{n}) + 2 a^{m} b^{n} a^{m} + 2 a^{m} b^{n} (- b^{n}) + b^{2 n} a^{m} + b^{2 n} (- b^{n})$

Step 5.1.2

Rewrite using the commutative property of multiplication.

$a^{3 m} - a^{2 m} b^{n} + 2 a^{m} b^{n} a^{m} + 2 a^{m} b^{n} (- b^{n}) + b^{2 n} a^{m} + b^{2 n} (- b^{n})$

Step 5.1.3

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

Tap for more steps...

Step 5.1.3.1

Move $a^{m}$ .

$a^{3 m} - a^{2 m} b^{n} + 2 (a^{m} a^{m}) b^{n} + 2 a^{m} b^{n} (- b^{n}) + b^{2 n} a^{m} + b^{2 n} (- b^{n})$

Step 5.1.3.2

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

$a^{3 m} - a^{2 m} b^{n} + 2 a^{m + m} b^{n} + 2 a^{m} b^{n} (- b^{n}) + b^{2 n} a^{m} + b^{2 n} (- b^{n})$

Step 5.1.3.3

Add $m$ and $m$ .

$a^{3 m} - a^{2 m} b^{n} + 2 a^{2 m} b^{n} + 2 a^{m} b^{n} (- b^{n}) + b^{2 n} a^{m} + b^{2 n} (- b^{n})$

Step 5.1.4

Multiply $b^{n}$ by $b^{n}$ by adding the exponents.

Tap for more steps...

Step 5.1.4.1

Move $b^{n}$ .

$a^{3 m} - a^{2 m} b^{n} + 2 a^{2 m} b^{n} + 2 a^{m} (b^{n} b^{n}) \cdot - 1 + b^{2 n} a^{m} + b^{2 n} (- b^{n})$

Step 5.1.4.2

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

$a^{3 m} - a^{2 m} b^{n} + 2 a^{2 m} b^{n} + 2 a^{m} b^{n + n} \cdot - 1 + b^{2 n} a^{m} + b^{2 n} (- b^{n})$

Step 5.1.4.3

Add $n$ and $n$ .

$a^{3 m} - a^{2 m} b^{n} + 2 a^{2 m} b^{n} + 2 a^{m} b^{2 n} \cdot - 1 + b^{2 n} a^{m} + b^{2 n} (- b^{n})$

Step 5.1.5

Multiply $- 1$ by $2$ .

$a^{3 m} - a^{2 m} b^{n} + 2 a^{2 m} b^{n} - 2 a^{m} b^{2 n} + b^{2 n} a^{m} + b^{2 n} (- b^{n})$

Step 5.1.6

Rewrite using the commutative property of multiplication.

$a^{3 m} - a^{2 m} b^{n} + 2 a^{2 m} b^{n} - 2 a^{m} b^{2 n} + b^{2 n} a^{m} - b^{2 n} b^{n}$

Step 5.1.7

Multiply $b^{2 n}$ by $b^{n}$ by adding the exponents.

Tap for more steps...

Step 5.1.7.1

Move $b^{n}$ .

$a^{3 m} - a^{2 m} b^{n} + 2 a^{2 m} b^{n} - 2 a^{m} b^{2 n} + b^{2 n} a^{m} - (b^{n} b^{2 n})$

Step 5.1.7.2

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

$a^{3 m} - a^{2 m} b^{n} + 2 a^{2 m} b^{n} - 2 a^{m} b^{2 n} + b^{2 n} a^{m} - b^{n + 2 n}$

Step 5.1.7.3

Add $n$ and $2 n$ .

$a^{3 m} - a^{2 m} b^{n} + 2 a^{2 m} b^{n} - 2 a^{m} b^{2 n} + b^{2 n} a^{m} - b^{3 n}$

Step 5.2

Simplify by adding terms.

Tap for more steps...

Step 5.2.1

Add $- a^{2 m} b^{n}$ and $2 a^{2 m} b^{n}$ .

$a^{3 m} + 1 a^{2 m} b^{n} - 2 a^{m} b^{2 n} + b^{2 n} a^{m} - b^{3 n}$

Step 5.2.2

Multiply $a^{2 m}$ by $1$ .

$a^{3 m} + a^{2 m} b^{n} - 2 a^{m} b^{2 n} + b^{2 n} a^{m} - b^{3 n}$

Step 6

Add $- 2 a^{m} b^{2 n}$ and $b^{2 n} a^{m}$ .

Tap for more steps...

Step 6.1

Reorder $b^{2 n}$ and $a^{m}$ .

$a^{3 m} + a^{2 m} b^{n} - 2 a^{m} b^{2 n} + a^{m} b^{2 n} - b^{3 n}$

Step 6.2

Add $- 2 a^{m} b^{2 n}$ and $a^{m} b^{2 n}$ .

$a^{3 m} + a^{2 m} b^{n} - a^{m} b^{2 n} - b^{3 n}$