Pre-Algebra Examples

$\frac{12 k^{2} - 5 k - 2}{k^{2} + 9 k} \div \frac{3 k^{2}}{k^{3} + 7 k^{2} - 18 k}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{12 k^{2} - 5 k - 2}{k^{2} + 9 k} \cdot \frac{k^{3} + 7 k^{2} - 18 k}{3 k^{2}}$

Step 2

Factor by grouping.

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Step 2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 12 \cdot - 2 = - 24$ and whose sum is $b = - 5$ .

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Step 2.1.1

Factor $- 5$ out of $- 5 k$ .

$\frac{12 k^{2} - 5 (k) - 2}{k^{2} + 9 k} \cdot \frac{k^{3} + 7 k^{2} - 18 k}{3 k^{2}}$

Step 2.1.2

Rewrite $- 5$ as $3$ plus $- 8$

$\frac{12 k^{2} + (3 - 8) k - 2}{k^{2} + 9 k} \cdot \frac{k^{3} + 7 k^{2} - 18 k}{3 k^{2}}$

Step 2.1.3

Apply the distributive property.

$\frac{12 k^{2} + 3 k - 8 k - 2}{k^{2} + 9 k} \cdot \frac{k^{3} + 7 k^{2} - 18 k}{3 k^{2}}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(12 k^{2} + 3 k) - 8 k - 2}{k^{2} + 9 k} \cdot \frac{k^{3} + 7 k^{2} - 18 k}{3 k^{2}}$

Step 2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{3 k (4 k + 1) - 2 (4 k + 1)}{k^{2} + 9 k} \cdot \frac{k^{3} + 7 k^{2} - 18 k}{3 k^{2}}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $4 k + 1$ .

$\frac{(4 k + 1) (3 k - 2)}{k^{2} + 9 k} \cdot \frac{k^{3} + 7 k^{2} - 18 k}{3 k^{2}}$

Step 3

Factor $k$ out of $k^{2} + 9 k$ .

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Step 3.1

Factor $k$ out of $k^{2}$ .

$\frac{(4 k + 1) (3 k - 2)}{k \cdot k + 9 k} \cdot \frac{k^{3} + 7 k^{2} - 18 k}{3 k^{2}}$

Step 3.2

Factor $k$ out of $9 k$ .

$\frac{(4 k + 1) (3 k - 2)}{k \cdot k + k \cdot 9} \cdot \frac{k^{3} + 7 k^{2} - 18 k}{3 k^{2}}$

Step 3.3

Factor $k$ out of $k \cdot k + k \cdot 9$ .

$\frac{(4 k + 1) (3 k - 2)}{k (k + 9)} \cdot \frac{k^{3} + 7 k^{2} - 18 k}{3 k^{2}}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $k$ out of $k^{3} + 7 k^{2} - 18 k$ .

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Step 4.1.1

Factor $k$ out of $k^{3}$ .

$\frac{(4 k + 1) (3 k - 2)}{k (k + 9)} \cdot \frac{k \cdot k^{2} + 7 k^{2} - 18 k}{3 k^{2}}$

Step 4.1.2

Factor $k$ out of $7 k^{2}$ .

$\frac{(4 k + 1) (3 k - 2)}{k (k + 9)} \cdot \frac{k \cdot k^{2} + k (7 k) - 18 k}{3 k^{2}}$

Step 4.1.3

Factor $k$ out of $- 18 k$ .

$\frac{(4 k + 1) (3 k - 2)}{k (k + 9)} \cdot \frac{k \cdot k^{2} + k (7 k) + k \cdot - 18}{3 k^{2}}$

Step 4.1.4

Factor $k$ out of $k \cdot k^{2} + k (7 k)$ .

$\frac{(4 k + 1) (3 k - 2)}{k (k + 9)} \cdot \frac{k (k^{2} + 7 k) + k \cdot - 18}{3 k^{2}}$

Step 4.1.5

Factor $k$ out of $k (k^{2} + 7 k) + k \cdot - 18$ .

$\frac{(4 k + 1) (3 k - 2)}{k (k + 9)} \cdot \frac{k (k^{2} + 7 k - 18)}{3 k^{2}}$

Step 4.2

Factor $k^{2} + 7 k - 18$ using the AC method.

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Step 4.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 18$ and whose sum is $7$ .

$- 2, 9$

Step 4.2.2

Write the factored form using these integers.

$\frac{(4 k + 1) (3 k - 2)}{k (k + 9)} \cdot \frac{k ((k - 2) (k + 9))}{3 k^{2}}$

$\frac{(4 k + 1) (3 k - 2)}{k (k + 9)} \cdot \frac{k (k - 2) (k + 9)}{3 k^{2}}$

Step 5

Combine.

$\frac{(4 k + 1) (3 k - 2) (k (k - 2) (k + 9))}{k (k + 9) (3 k^{2})}$

Step 6

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

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Step 6.1

Move $k^{2}$ .

$\frac{(4 k + 1) (3 k - 2) (k (k - 2) (k + 9))}{k^{2} k (k + 9) \cdot 3}$

Step 6.2

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

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Step 6.2.1

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

$\frac{(4 k + 1) (3 k - 2) (k (k - 2) (k + 9))}{k^{2} k^{1} (k + 9) \cdot 3}$

Step 6.2.2

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

$\frac{(4 k + 1) (3 k - 2) (k (k - 2) (k + 9))}{k^{2 + 1} (k + 9) \cdot 3}$

Step 6.3

Add $2$ and $1$ .

$\frac{(4 k + 1) (3 k - 2) (k (k - 2) (k + 9))}{k^{3} (k + 9) \cdot 3}$

Step 7

Cancel the common factor of $k$ and $k^{3}$ .

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Step 7.1

Factor $k$ out of $(4 k + 1) (3 k - 2) (k (k - 2) (k + 9))$ .

$\frac{k ((4 k + 1) (3 k - 2) ((k - 2) (k + 9)))}{k^{3} (k + 9) \cdot 3}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $k$ out of $k^{3} (k + 9) \cdot 3$ .

$\frac{k ((4 k + 1) (3 k - 2) ((k - 2) (k + 9)))}{k ((k^{2} (k + 9)) \cdot 3)}$

Step 7.2.2

Cancel the common factor.

$\frac{k ((4 k + 1) (3 k - 2) ((k - 2) (k + 9)))}{k ((k^{2} (k + 9)) \cdot 3)}$

Step 7.2.3

Rewrite the expression.

$\frac{(4 k + 1) (3 k - 2) ((k - 2) (k + 9))}{(k^{2} (k + 9)) \cdot 3}$

Step 8

Cancel the common factor of $k + 9$ .

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Step 8.1

Cancel the common factor.

$\frac{(4 k + 1) (3 k - 2) ((k - 2) (k + 9))}{k^{2} (k + 9) \cdot 3}$

Step 8.2

Rewrite the expression.

$\frac{(4 k + 1) (3 k - 2) (k - 2)}{(k^{2}) \cdot 3}$

Step 9

Move $3$ to the left of $k^{2}$ .

$\frac{(4 k + 1) (3 k - 2) (k - 2)}{3 k^{2}}$

Step 10

Expand $(4 k + 1) (3 k - 2)$ using the FOIL Method.

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Step 10.1

Apply the distributive property.

$\frac{(4 k (3 k - 2) + 1 (3 k - 2)) (k - 2)}{3 k^{2}}$

Step 10.2

Apply the distributive property.

$\frac{(4 k (3 k) + 4 k \cdot - 2 + 1 (3 k - 2)) (k - 2)}{3 k^{2}}$

Step 10.3

Apply the distributive property.

$\frac{(4 k (3 k) + 4 k \cdot - 2 + 1 (3 k) + 1 \cdot - 2) (k - 2)}{3 k^{2}}$

Step 11

Simplify and combine like terms.

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Step 11.1

Simplify each term.

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Step 11.1.1

Rewrite using the commutative property of multiplication.

$\frac{(4 \cdot 3 k \cdot k + 4 k \cdot - 2 + 1 (3 k) + 1 \cdot - 2) (k - 2)}{3 k^{2}}$

Step 11.1.2

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

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Step 11.1.2.1

Move $k$ .

$\frac{(4 \cdot 3 (k \cdot k) + 4 k \cdot - 2 + 1 (3 k) + 1 \cdot - 2) (k - 2)}{3 k^{2}}$

Step 11.1.2.2

Multiply $k$ by $k$ .

$\frac{(4 \cdot 3 k^{2} + 4 k \cdot - 2 + 1 (3 k) + 1 \cdot - 2) (k - 2)}{3 k^{2}}$

Step 11.1.3

Multiply $4$ by $3$ .

$\frac{(12 k^{2} + 4 k \cdot - 2 + 1 (3 k) + 1 \cdot - 2) (k - 2)}{3 k^{2}}$

Step 11.1.4

Multiply $- 2$ by $4$ .

$\frac{(12 k^{2} - 8 k + 1 (3 k) + 1 \cdot - 2) (k - 2)}{3 k^{2}}$

Step 11.1.5

Multiply $3 k$ by $1$ .

$\frac{(12 k^{2} - 8 k + 3 k + 1 \cdot - 2) (k - 2)}{3 k^{2}}$

Step 11.1.6

Multiply $- 2$ by $1$ .

$\frac{(12 k^{2} - 8 k + 3 k - 2) (k - 2)}{3 k^{2}}$

Step 11.2

Add $- 8 k$ and $3 k$ .

$\frac{(12 k^{2} - 5 k - 2) (k - 2)}{3 k^{2}}$

Step 12

Expand $(12 k^{2} - 5 k - 2) (k - 2)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{12 k^{2} k + 12 k^{2} \cdot - 2 - 5 k \cdot k - 5 k \cdot - 2 - 2 k - 2 \cdot - 2}{3 k^{2}}$

Step 13

Simplify each term.

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Step 13.1

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

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Step 13.1.1

Move $k$ .

$\frac{12 (k \cdot k^{2}) + 12 k^{2} \cdot - 2 - 5 k \cdot k - 5 k \cdot - 2 - 2 k - 2 \cdot - 2}{3 k^{2}}$

Step 13.1.2

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

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Step 13.1.2.1

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

$\frac{12 (k^{1} k^{2}) + 12 k^{2} \cdot - 2 - 5 k \cdot k - 5 k \cdot - 2 - 2 k - 2 \cdot - 2}{3 k^{2}}$

Step 13.1.2.2

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

$\frac{12 k^{1 + 2} + 12 k^{2} \cdot - 2 - 5 k \cdot k - 5 k \cdot - 2 - 2 k - 2 \cdot - 2}{3 k^{2}}$

Step 13.1.3

Add $1$ and $2$ .

$\frac{12 k^{3} + 12 k^{2} \cdot - 2 - 5 k \cdot k - 5 k \cdot - 2 - 2 k - 2 \cdot - 2}{3 k^{2}}$

Step 13.2

Multiply $- 2$ by $12$ .

$\frac{12 k^{3} - 24 k^{2} - 5 k \cdot k - 5 k \cdot - 2 - 2 k - 2 \cdot - 2}{3 k^{2}}$

Step 13.3

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

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Step 13.3.1

Move $k$ .

$\frac{12 k^{3} - 24 k^{2} - 5 (k \cdot k) - 5 k \cdot - 2 - 2 k - 2 \cdot - 2}{3 k^{2}}$

Step 13.3.2

Multiply $k$ by $k$ .

$\frac{12 k^{3} - 24 k^{2} - 5 k^{2} - 5 k \cdot - 2 - 2 k - 2 \cdot - 2}{3 k^{2}}$

Step 13.4

Multiply $- 2$ by $- 5$ .

$\frac{12 k^{3} - 24 k^{2} - 5 k^{2} + 10 k - 2 k - 2 \cdot - 2}{3 k^{2}}$

Step 13.5

Multiply $- 2$ by $- 2$ .

$\frac{12 k^{3} - 24 k^{2} - 5 k^{2} + 10 k - 2 k + 4}{3 k^{2}}$

Step 14

Subtract $5 k^{2}$ from $- 24 k^{2}$ .

$\frac{12 k^{3} - 29 k^{2} + 10 k - 2 k + 4}{3 k^{2}}$

Step 15

Subtract $2 k$ from $10 k$ .

$\frac{12 k^{3} - 29 k^{2} + 8 k + 4}{3 k^{2}}$

Step 16

Split the fraction $\frac{12 k^{3} - 29 k^{2} + 8 k + 4}{3 k^{2}}$ into two fractions.

$\frac{12 k^{3} - 29 k^{2} + 8 k}{3 k^{2}} + \frac{4}{3 k^{2}}$

Step 17

Split the fraction $\frac{12 k^{3} - 29 k^{2} + 8 k}{3 k^{2}}$ into two fractions.

$\frac{12 k^{3} - 29 k^{2}}{3 k^{2}} + \frac{8 k}{3 k^{2}} + \frac{4}{3 k^{2}}$

Step 18

Split the fraction $\frac{12 k^{3} - 29 k^{2}}{3 k^{2}}$ into two fractions.

$\frac{12 k^{3}}{3 k^{2}} + \frac{- 29 k^{2}}{3 k^{2}} + \frac{8 k}{3 k^{2}} + \frac{4}{3 k^{2}}$

Step 19

Cancel the common factor of $12$ and $3$ .

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Step 19.1

Factor $3$ out of $12 k^{3}$ .

$\frac{3 (4 k^{3})}{3 k^{2}} + \frac{- 29 k^{2}}{3 k^{2}} + \frac{8 k}{3 k^{2}} + \frac{4}{3 k^{2}}$

Step 19.2

Cancel the common factors.

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Step 19.2.1

Factor $3$ out of $3 k^{2}$ .

$\frac{3 (4 k^{3})}{3 (k^{2})} + \frac{- 29 k^{2}}{3 k^{2}} + \frac{8 k}{3 k^{2}} + \frac{4}{3 k^{2}}$

Step 19.2.2

Cancel the common factor.

$\frac{3 (4 k^{3})}{3 k^{2}} + \frac{- 29 k^{2}}{3 k^{2}} + \frac{8 k}{3 k^{2}} + \frac{4}{3 k^{2}}$

Step 19.2.3

Rewrite the expression.

$\frac{4 k^{3}}{k^{2}} + \frac{- 29 k^{2}}{3 k^{2}} + \frac{8 k}{3 k^{2}} + \frac{4}{3 k^{2}}$

Step 20

Cancel the common factor of $k^{3}$ and $k^{2}$ .

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Step 20.1

Factor $k^{2}$ out of $4 k^{3}$ .

$\frac{k^{2} (4 k)}{k^{2}} + \frac{- 29 k^{2}}{3 k^{2}} + \frac{8 k}{3 k^{2}} + \frac{4}{3 k^{2}}$

Step 20.2

Cancel the common factors.

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Step 20.2.1

Multiply by $1$ .

$\frac{k^{2} (4 k)}{k^{2} \cdot 1} + \frac{- 29 k^{2}}{3 k^{2}} + \frac{8 k}{3 k^{2}} + \frac{4}{3 k^{2}}$

Step 20.2.2

Cancel the common factor.

$\frac{k^{2} (4 k)}{k^{2} \cdot 1} + \frac{- 29 k^{2}}{3 k^{2}} + \frac{8 k}{3 k^{2}} + \frac{4}{3 k^{2}}$

Step 20.2.3

Rewrite the expression.

$\frac{4 k}{1} + \frac{- 29 k^{2}}{3 k^{2}} + \frac{8 k}{3 k^{2}} + \frac{4}{3 k^{2}}$

Step 20.2.4

Divide $4 k$ by $1$ .

$4 k + \frac{- 29 k^{2}}{3 k^{2}} + \frac{8 k}{3 k^{2}} + \frac{4}{3 k^{2}}$

Step 21

Cancel the common factor of $k^{2}$ .

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Step 21.1

Cancel the common factor.

$4 k + \frac{- 29 k^{2}}{3 k^{2}} + \frac{8 k}{3 k^{2}} + \frac{4}{3 k^{2}}$

Step 21.2

Rewrite the expression.

$4 k + \frac{- 29}{3} + \frac{8 k}{3 k^{2}} + \frac{4}{3 k^{2}}$

Step 22

Move the negative in front of the fraction.

$4 k - \frac{29}{3} + \frac{8 k}{3 k^{2}} + \frac{4}{3 k^{2}}$

Step 23

Cancel the common factor of $k$ and $k^{2}$ .

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Step 23.1

Factor $k$ out of $8 k$ .

$4 k - \frac{29}{3} + \frac{k \cdot 8}{3 k^{2}} + \frac{4}{3 k^{2}}$

Step 23.2

Cancel the common factors.

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Step 23.2.1

Factor $k$ out of $3 k^{2}$ .

$4 k - \frac{29}{3} + \frac{k \cdot 8}{k (3 k)} + \frac{4}{3 k^{2}}$

Step 23.2.2

Cancel the common factor.

$4 k - \frac{29}{3} + \frac{k \cdot 8}{k (3 k)} + \frac{4}{3 k^{2}}$

Step 23.2.3

Rewrite the expression.

$4 k - \frac{29}{3} + \frac{8}{3 k} + \frac{4}{3 k^{2}}$