Basic Math Examples

${(m + n)}^{3} - {(m - n)}^{3}$

Step 1

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = m + n$ and $b = m - n$ .

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

Step 2

Simplify.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Multiply $- - n$ .

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Step 2.2.1

Multiply $- 1$ by $- 1$ .

$(m + n - m + 1 n) ({(m + n)}^{2} + (m + n) (m - n) + {(m - n)}^{2})$

Step 2.2.2

Multiply $n$ by $1$ .

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

Step 2.3

Subtract $m$ from $m$ .

$(n + 0 + n) ({(m + n)}^{2} + (m + n) (m - n) + {(m - n)}^{2})$

Step 2.4

Add $n$ and $0$ .

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

Step 2.5

Add $n$ and $n$ .

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

Step 2.6

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

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

Step 2.7

Expand $(m + n) (m + n)$ using the FOIL Method.

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Step 2.7.1

Apply the distributive property.

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

Step 2.7.2

Apply the distributive property.

$2 n (m \cdot m + m n + n (m + n) + (m + n) (m - n) + {(m - n)}^{2})$

Step 2.7.3

Apply the distributive property.

$2 n (m \cdot m + m n + n m + n \cdot n + (m + n) (m - n) + {(m - n)}^{2})$

Step 2.8

Simplify and combine like terms.

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

Multiply $m$ by $m$ .

$2 n (m^{2} + m n + n m + n \cdot n + (m + n) (m - n) + {(m - n)}^{2})$

Step 2.8.1.2

Multiply $n$ by $n$ .

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

Step 2.8.2

Add $m n$ and $n m$ .

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Step 2.8.2.1

Reorder $n$ and $m$ .

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

Step 2.8.2.2

Add $m n$ and $m n$ .

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

Step 2.9

Expand $(m + n) (m - n)$ using the FOIL Method.

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Step 2.9.1

Apply the distributive property.

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

Step 2.9.2

Apply the distributive property.

$2 n (m^{2} + 2 m n + n^{2} + m \cdot m + m (- n) + n (m - n) + {(m - n)}^{2})$

Step 2.9.3

Apply the distributive property.

$2 n (m^{2} + 2 m n + n^{2} + m \cdot m + m (- n) + n m + n (- n) + {(m - n)}^{2})$

Step 2.10

Combine the opposite terms in $m \cdot m + m (- n) + n m + n (- n)$ .

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Step 2.10.1

Reorder the factors in the terms $m (- n)$ and $n m$ .

$2 n (m^{2} + 2 m n + n^{2} + m \cdot m - m n + m n + n (- n) + {(m - n)}^{2})$

Step 2.10.2

Add $- m n$ and $m n$ .

$2 n (m^{2} + 2 m n + n^{2} + m \cdot m + 0 + n (- n) + {(m - n)}^{2})$

Step 2.10.3

Add $m \cdot m$ and $0$ .

$2 n (m^{2} + 2 m n + n^{2} + m \cdot m + n (- n) + {(m - n)}^{2})$

Step 2.11

Simplify each term.

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Step 2.11.1

Multiply $m$ by $m$ .

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

Step 2.11.2

Rewrite using the commutative property of multiplication.

$2 n (m^{2} + 2 m n + n^{2} + m^{2} - n \cdot n + {(m - n)}^{2})$

Step 2.11.3

Multiply $n$ by $n$ by adding the exponents.

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Step 2.11.3.1

Move $n$ .

$2 n (m^{2} + 2 m n + n^{2} + m^{2} - (n \cdot n) + {(m - n)}^{2})$

Step 2.11.3.2

Multiply $n$ by $n$ .

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

Step 2.12

Rewrite ${(m - n)}^{2}$ as $(m - n) (m - n)$ .

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

Step 2.13

Expand $(m - n) (m - n)$ using the FOIL Method.

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Step 2.13.1

Apply the distributive property.

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

Step 2.13.2

Apply the distributive property.

$2 n (m^{2} + 2 m n + n^{2} + m^{2} - n^{2} + m \cdot m + m (- n) - n (m - n))$

Step 2.13.3

Apply the distributive property.

$2 n (m^{2} + 2 m n + n^{2} + m^{2} - n^{2} + m \cdot m + m (- n) - n m - n (- n))$

Step 2.14

Simplify and combine like terms.

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Step 2.14.1

Simplify each term.

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Step 2.14.1.1

Multiply $m$ by $m$ .

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

Step 2.14.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.14.1.3

Rewrite using the commutative property of multiplication.

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

Step 2.14.1.4

Multiply $n$ by $n$ by adding the exponents.

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Step 2.14.1.4.1

Move $n$ .

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

Step 2.14.1.4.2

Multiply $n$ by $n$ .

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

Step 2.14.1.5

Multiply $- 1$ by $- 1$ .

$2 n (m^{2} + 2 m n + n^{2} + m^{2} - n^{2} + m^{2} - m n - n m + 1 n^{2})$

Step 2.14.1.6

Multiply $n^{2}$ by $1$ .

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

Step 2.14.2

Subtract $n m$ from $- m n$ .

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Step 2.14.2.1

Move $n$ .

$2 n (m^{2} + 2 m n + n^{2} + m^{2} - n^{2} + m^{2} - m n - 1 m n + n^{2})$

Step 2.14.2.2

Subtract $m n$ from $- m n$ .

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

Step 2.15

Add $m^{2}$ and $m^{2}$ .

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

Step 2.16

Add $2 m^{2}$ and $m^{2}$ .

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

Step 2.17

Subtract $2 m n$ from $2 m n$ .

$2 n (3 m^{2} + n^{2} - n^{2} + 0 + n^{2})$

Step 2.18

Add $3 m^{2}$ and $0$ .

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

Step 2.19

Subtract $n^{2}$ from $n^{2}$ .

$2 n (3 m^{2} + 0 + n^{2})$

Step 2.20

Add $3 m^{2}$ and $0$ .

$2 n (3 m^{2} + n^{2})$