Algebra Examples

$(6 m^{2} - 8 m n + 4 n^{2}) (8 m + 8 n)$

Step 1

Expand $(6 m^{2} - 8 m n + 4 n^{2}) (8 m + 8 n)$ by multiplying each term in the first expression by each term in the second expression.

$6 m^{2} (8 m) + 6 m^{2} (8 n) - 8 m n (8 m) - 8 m n (8 n) + 4 n^{2} (8 m) + 4 n^{2} (8 n)$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

$6 \cdot 8 m^{2} m + 6 m^{2} (8 n) - 8 m n (8 m) - 8 m n (8 n) + 4 n^{2} (8 m) + 4 n^{2} (8 n)$

Step 2.1.2

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

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Step 2.1.2.1

Move $m$ .

$6 \cdot 8 (m \cdot m^{2}) + 6 m^{2} (8 n) - 8 m n (8 m) - 8 m n (8 n) + 4 n^{2} (8 m) + 4 n^{2} (8 n)$

Step 2.1.2.2

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

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Step 2.1.2.2.1

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

$6 \cdot 8 (m^{1} m^{2}) + 6 m^{2} (8 n) - 8 m n (8 m) - 8 m n (8 n) + 4 n^{2} (8 m) + 4 n^{2} (8 n)$

Step 2.1.2.2.2

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

$6 \cdot 8 m^{1 + 2} + 6 m^{2} (8 n) - 8 m n (8 m) - 8 m n (8 n) + 4 n^{2} (8 m) + 4 n^{2} (8 n)$

Step 2.1.2.3

Add $1$ and $2$ .

$6 \cdot 8 m^{3} + 6 m^{2} (8 n) - 8 m n (8 m) - 8 m n (8 n) + 4 n^{2} (8 m) + 4 n^{2} (8 n)$

Step 2.1.3

Multiply $6$ by $8$ .

$48 m^{3} + 6 m^{2} (8 n) - 8 m n (8 m) - 8 m n (8 n) + 4 n^{2} (8 m) + 4 n^{2} (8 n)$

Step 2.1.4

Rewrite using the commutative property of multiplication.

$48 m^{3} + 6 \cdot 8 m^{2} n - 8 m n (8 m) - 8 m n (8 n) + 4 n^{2} (8 m) + 4 n^{2} (8 n)$

Step 2.1.5

Multiply $6$ by $8$ .

$48 m^{3} + 48 m^{2} n - 8 m n (8 m) - 8 m n (8 n) + 4 n^{2} (8 m) + 4 n^{2} (8 n)$

Step 2.1.6

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

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Step 2.1.6.1

Move $m$ .

$48 m^{3} + 48 m^{2} n - 8 (m \cdot m) n \cdot 8 - 8 m n (8 n) + 4 n^{2} (8 m) + 4 n^{2} (8 n)$

Step 2.1.6.2

Multiply $m$ by $m$ .

$48 m^{3} + 48 m^{2} n - 8 m^{2} n \cdot 8 - 8 m n (8 n) + 4 n^{2} (8 m) + 4 n^{2} (8 n)$

Step 2.1.7

Multiply $8$ by $- 8$ .

$48 m^{3} + 48 m^{2} n - 64 m^{2} n - 8 m n (8 n) + 4 n^{2} (8 m) + 4 n^{2} (8 n)$

Step 2.1.8

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

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Step 2.1.8.1

Move $n$ .

$48 m^{3} + 48 m^{2} n - 64 m^{2} n - 8 m (n \cdot n) \cdot 8 + 4 n^{2} (8 m) + 4 n^{2} (8 n)$

Step 2.1.8.2

Multiply $n$ by $n$ .

$48 m^{3} + 48 m^{2} n - 64 m^{2} n - 8 m n^{2} \cdot 8 + 4 n^{2} (8 m) + 4 n^{2} (8 n)$

Step 2.1.9

Multiply $8$ by $- 8$ .

$48 m^{3} + 48 m^{2} n - 64 m^{2} n - 64 m n^{2} + 4 n^{2} (8 m) + 4 n^{2} (8 n)$

Step 2.1.10

Rewrite using the commutative property of multiplication.

$48 m^{3} + 48 m^{2} n - 64 m^{2} n - 64 m n^{2} + 4 \cdot 8 n^{2} m + 4 n^{2} (8 n)$

Step 2.1.11

Multiply $4$ by $8$ .

$48 m^{3} + 48 m^{2} n - 64 m^{2} n - 64 m n^{2} + 32 n^{2} m + 4 n^{2} (8 n)$

Step 2.1.12

Rewrite using the commutative property of multiplication.

$48 m^{3} + 48 m^{2} n - 64 m^{2} n - 64 m n^{2} + 32 n^{2} m + 4 \cdot 8 n^{2} n$

Step 2.1.13

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

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Step 2.1.13.1

Move $n$ .

$48 m^{3} + 48 m^{2} n - 64 m^{2} n - 64 m n^{2} + 32 n^{2} m + 4 \cdot 8 (n \cdot n^{2})$

Step 2.1.13.2

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

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Step 2.1.13.2.1

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

$48 m^{3} + 48 m^{2} n - 64 m^{2} n - 64 m n^{2} + 32 n^{2} m + 4 \cdot 8 (n^{1} n^{2})$

Step 2.1.13.2.2

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

$48 m^{3} + 48 m^{2} n - 64 m^{2} n - 64 m n^{2} + 32 n^{2} m + 4 \cdot 8 n^{1 + 2}$

Step 2.1.13.3

Add $1$ and $2$ .

$48 m^{3} + 48 m^{2} n - 64 m^{2} n - 64 m n^{2} + 32 n^{2} m + 4 \cdot 8 n^{3}$

Step 2.1.14

Multiply $4$ by $8$ .

$48 m^{3} + 48 m^{2} n - 64 m^{2} n - 64 m n^{2} + 32 n^{2} m + 32 n^{3}$

Step 2.2

Subtract $64 m^{2} n$ from $48 m^{2} n$ .

$48 m^{3} - 16 m^{2} n - 64 m n^{2} + 32 n^{2} m + 32 n^{3}$

Step 3

Add $- 64 m n^{2}$ and $32 n^{2} m$ .

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Step 3.1

Move $n^{2}$ .

$48 m^{3} - 16 m^{2} n - 64 m n^{2} + 32 m n^{2} + 32 n^{3}$

Step 3.2

Add $- 64 m n^{2}$ and $32 m n^{2}$ .

$48 m^{3} - 16 m^{2} n - 32 m n^{2} + 32 n^{3}$