Basic Math Examples

$(m - 5 n) (m^{2} + 5 m n + 25 n^{2})$

Step 1

Expand $(m - 5 n) (m^{2} + 5 m n + 25 n^{2})$ by multiplying each term in the first expression by each term in the second expression.

$m \cdot m^{2} + m (5 m n) + m (25 n^{2}) - 5 n m^{2} - 5 n (5 m n) - 5 n (25 n^{2})$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

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

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Step 2.1.1.1.1

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

$m^{1} m^{2} + m (5 m n) + m (25 n^{2}) - 5 n m^{2} - 5 n (5 m n) - 5 n (25 n^{2})$

Step 2.1.1.1.2

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

$m^{1 + 2} + m (5 m n) + m (25 n^{2}) - 5 n m^{2} - 5 n (5 m n) - 5 n (25 n^{2})$

Step 2.1.1.2

Add $1$ and $2$ .

$m^{3} + m (5 m n) + m (25 n^{2}) - 5 n m^{2} - 5 n (5 m n) - 5 n (25 n^{2})$

Step 2.1.2

Rewrite using the commutative property of multiplication.

$m^{3} + 5 m (m n) + m (25 n^{2}) - 5 n m^{2} - 5 n (5 m n) - 5 n (25 n^{2})$

Step 2.1.3

Multiply $m$ by $m$ by adding the exponents.

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Step 2.1.3.1

Move $m$ .

$m^{3} + 5 (m \cdot m) n + m (25 n^{2}) - 5 n m^{2} - 5 n (5 m n) - 5 n (25 n^{2})$

Step 2.1.3.2

Multiply $m$ by $m$ .

$m^{3} + 5 m^{2} n + m (25 n^{2}) - 5 n m^{2} - 5 n (5 m n) - 5 n (25 n^{2})$

Step 2.1.4

Rewrite using the commutative property of multiplication.

$m^{3} + 5 m^{2} n + 25 m n^{2} - 5 n m^{2} - 5 n (5 m n) - 5 n (25 n^{2})$

Step 2.1.5

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

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Step 2.1.5.1

Move $n$ .

$m^{3} + 5 m^{2} n + 25 m n^{2} - 5 n m^{2} - 5 (n \cdot n) (5 m) - 5 n (25 n^{2})$

Step 2.1.5.2

Multiply $n$ by $n$ .

$m^{3} + 5 m^{2} n + 25 m n^{2} - 5 n m^{2} - 5 n^{2} (5 m) - 5 n (25 n^{2})$

Step 2.1.6

Rewrite using the commutative property of multiplication.

$m^{3} + 5 m^{2} n + 25 m n^{2} - 5 n m^{2} - 5 \cdot 5 n^{2} m - 5 n (25 n^{2})$

Step 2.1.7

Multiply $- 5$ by $5$ .

$m^{3} + 5 m^{2} n + 25 m n^{2} - 5 n m^{2} - 25 n^{2} m - 5 n (25 n^{2})$

Step 2.1.8

Rewrite using the commutative property of multiplication.

$m^{3} + 5 m^{2} n + 25 m n^{2} - 5 n m^{2} - 25 n^{2} m - 5 \cdot 25 n \cdot n^{2}$

Step 2.1.9

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

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Step 2.1.9.1

Move $n^{2}$ .

$m^{3} + 5 m^{2} n + 25 m n^{2} - 5 n m^{2} - 25 n^{2} m - 5 \cdot 25 (n^{2} n)$

Step 2.1.9.2

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

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Step 2.1.9.2.1

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

$m^{3} + 5 m^{2} n + 25 m n^{2} - 5 n m^{2} - 25 n^{2} m - 5 \cdot 25 (n^{2} n^{1})$

Step 2.1.9.2.2

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

$m^{3} + 5 m^{2} n + 25 m n^{2} - 5 n m^{2} - 25 n^{2} m - 5 \cdot 25 n^{2 + 1}$

Step 2.1.9.3

Add $2$ and $1$ .

$m^{3} + 5 m^{2} n + 25 m n^{2} - 5 n m^{2} - 25 n^{2} m - 5 \cdot 25 n^{3}$

Step 2.1.10

Multiply $- 5$ by $25$ .

$m^{3} + 5 m^{2} n + 25 m n^{2} - 5 n m^{2} - 25 n^{2} m - 125 n^{3}$

Step 2.2

Combine the opposite terms in $m^{3} + 5 m^{2} n + 25 m n^{2} - 5 n m^{2} - 25 n^{2} m - 125 n^{3}$ .

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Step 2.2.1

Reorder the factors in the terms $5 m^{2} n$ and $- 5 n m^{2}$ .

$m^{3} + 5 m^{2} n + 25 m n^{2} - 5 m^{2} n - 25 n^{2} m - 125 n^{3}$

Step 2.2.2

Subtract $5 m^{2} n$ from $5 m^{2} n$ .

$m^{3} + 25 m n^{2} + 0 - 25 n^{2} m - 125 n^{3}$

Step 2.2.3

Add $m^{3} + 25 m n^{2}$ and $0$ .

$m^{3} + 25 m n^{2} - 25 n^{2} m - 125 n^{3}$

Step 2.2.4

Reorder the factors in the terms $25 m n^{2}$ and $- 25 n^{2} m$ .

$m^{3} + 25 n^{2} m - 25 n^{2} m - 125 n^{3}$

Step 2.2.5

Subtract $25 n^{2} m$ from $25 n^{2} m$ .

$m^{3} + 0 - 125 n^{3}$

Step 2.2.6

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

$m^{3} - 125 n^{3}$