Basic Math Examples

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

Step 1

Expand $(m - 5 p) (m^{2} - 2 m p + 3 p^{2})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2

Simplify each term.

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Step 2.1

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

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Step 2.1.1

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

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Step 2.1.1.1

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

$m^{1} m^{2} + m (- 2 m p) + m (3 p^{2}) - 5 p m^{2} - 5 p (- 2 m p) - 5 p (3 p^{2})$

Step 2.1.1.2

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

$m^{1 + 2} + m (- 2 m p) + m (3 p^{2}) - 5 p m^{2} - 5 p (- 2 m p) - 5 p (3 p^{2})$

Step 2.1.2

Add $1$ and $2$ .

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

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

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Step 2.3.1

Move $m$ .

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

Step 2.3.2

Multiply $m$ by $m$ .

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

Step 2.4

Rewrite using the commutative property of multiplication.

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

Step 2.5

Multiply $p$ by $p$ by adding the exponents.

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Step 2.5.1

Move $p$ .

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

Step 2.5.2

Multiply $p$ by $p$ .

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

Step 2.6

Rewrite using the commutative property of multiplication.

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

Step 2.7

Multiply $- 5$ by $- 2$ .

$m^{3} - 2 m^{2} p + 3 m p^{2} - 5 p m^{2} + 10 p^{2} m - 5 p (3 p^{2})$

Step 2.8

Rewrite using the commutative property of multiplication.

$m^{3} - 2 m^{2} p + 3 m p^{2} - 5 p m^{2} + 10 p^{2} m - 5 \cdot 3 p \cdot p^{2}$

Step 2.9

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

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Step 2.9.1

Move $p^{2}$ .

$m^{3} - 2 m^{2} p + 3 m p^{2} - 5 p m^{2} + 10 p^{2} m - 5 \cdot 3 (p^{2} p)$

Step 2.9.2

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

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Step 2.9.2.1

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

$m^{3} - 2 m^{2} p + 3 m p^{2} - 5 p m^{2} + 10 p^{2} m - 5 \cdot 3 (p^{2} p^{1})$

Step 2.9.2.2

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

$m^{3} - 2 m^{2} p + 3 m p^{2} - 5 p m^{2} + 10 p^{2} m - 5 \cdot 3 p^{2 + 1}$

Step 2.9.3

Add $2$ and $1$ .

$m^{3} - 2 m^{2} p + 3 m p^{2} - 5 p m^{2} + 10 p^{2} m - 5 \cdot 3 p^{3}$

Step 2.10

Multiply $- 5$ by $3$ .

$m^{3} - 2 m^{2} p + 3 m p^{2} - 5 p m^{2} + 10 p^{2} m - 15 p^{3}$

Step 3

Subtract $5 p m^{2}$ from $- 2 m^{2} p$ .

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Step 3.1

Move $p$ .

$m^{3} - 2 m^{2} p - 5 m^{2} p + 3 m p^{2} + 10 p^{2} m - 15 p^{3}$

Step 3.2

Subtract $5 m^{2} p$ from $- 2 m^{2} p$ .

$m^{3} - 7 m^{2} p + 3 m p^{2} + 10 p^{2} m - 15 p^{3}$

Step 4

Add $3 m p^{2}$ and $10 p^{2} m$ .

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Step 4.1

Move $p^{2}$ .

$m^{3} - 7 m^{2} p + 3 m p^{2} + 10 m p^{2} - 15 p^{3}$

Step 4.2

Add $3 m p^{2}$ and $10 m p^{2}$ .

$m^{3} - 7 m^{2} p + 13 m p^{2} - 15 p^{3}$