Finite Math Examples

$(k^{2} + 5 k + 6) (k^{2} + 7 k)$

Step 1

Expand $(k^{2} + 5 k + 6) (k^{2} + 7 k)$ by multiplying each term in the first expression by each term in the second expression.

$k^{2} k^{2} + k^{2} (7 k) + 5 k \cdot k^{2} + 5 k (7 k) + 6 k^{2} + 6 (7 k)$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

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

$k^{2 + 2} + k^{2} (7 k) + 5 k \cdot k^{2} + 5 k (7 k) + 6 k^{2} + 6 (7 k)$

Step 2.1.1.2

Add $2$ and $2$ .

$k^{4} + k^{2} (7 k) + 5 k \cdot k^{2} + 5 k (7 k) + 6 k^{2} + 6 (7 k)$

Step 2.1.2

Rewrite using the commutative property of multiplication.

$k^{4} + 7 k^{2} k + 5 k \cdot k^{2} + 5 k (7 k) + 6 k^{2} + 6 (7 k)$

Step 2.1.3

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

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Step 2.1.3.1

Move $k$ .

$k^{4} + 7 (k \cdot k^{2}) + 5 k \cdot k^{2} + 5 k (7 k) + 6 k^{2} + 6 (7 k)$

Step 2.1.3.2

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

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Step 2.1.3.2.1

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

$k^{4} + 7 (k^{1} k^{2}) + 5 k \cdot k^{2} + 5 k (7 k) + 6 k^{2} + 6 (7 k)$

Step 2.1.3.2.2

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

$k^{4} + 7 k^{1 + 2} + 5 k \cdot k^{2} + 5 k (7 k) + 6 k^{2} + 6 (7 k)$

Step 2.1.3.3

Add $1$ and $2$ .

$k^{4} + 7 k^{3} + 5 k \cdot k^{2} + 5 k (7 k) + 6 k^{2} + 6 (7 k)$

Step 2.1.4

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

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Step 2.1.4.1

Move $k^{2}$ .

$k^{4} + 7 k^{3} + 5 (k^{2} k) + 5 k (7 k) + 6 k^{2} + 6 (7 k)$

Step 2.1.4.2

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

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Step 2.1.4.2.1

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

$k^{4} + 7 k^{3} + 5 (k^{2} k^{1}) + 5 k (7 k) + 6 k^{2} + 6 (7 k)$

Step 2.1.4.2.2

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

$k^{4} + 7 k^{3} + 5 k^{2 + 1} + 5 k (7 k) + 6 k^{2} + 6 (7 k)$

Step 2.1.4.3

Add $2$ and $1$ .

$k^{4} + 7 k^{3} + 5 k^{3} + 5 k (7 k) + 6 k^{2} + 6 (7 k)$

Step 2.1.5

Rewrite using the commutative property of multiplication.

$k^{4} + 7 k^{3} + 5 k^{3} + 5 \cdot 7 k \cdot k + 6 k^{2} + 6 (7 k)$

Step 2.1.6

Multiply $k$ by $k$ by adding the exponents.

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Step 2.1.6.1

Move $k$ .

$k^{4} + 7 k^{3} + 5 k^{3} + 5 \cdot 7 (k \cdot k) + 6 k^{2} + 6 (7 k)$

Step 2.1.6.2

Multiply $k$ by $k$ .

$k^{4} + 7 k^{3} + 5 k^{3} + 5 \cdot 7 k^{2} + 6 k^{2} + 6 (7 k)$

Step 2.1.7

Multiply $5$ by $7$ .

$k^{4} + 7 k^{3} + 5 k^{3} + 35 k^{2} + 6 k^{2} + 6 (7 k)$

Step 2.1.8

Multiply $7$ by $6$ .

$k^{4} + 7 k^{3} + 5 k^{3} + 35 k^{2} + 6 k^{2} + 42 k$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $7 k^{3}$ and $5 k^{3}$ .

$k^{4} + 12 k^{3} + 35 k^{2} + 6 k^{2} + 42 k$

Step 2.2.2

Add $35 k^{2}$ and $6 k^{2}$ .

$k^{4} + 12 k^{3} + 41 k^{2} + 42 k$