Pre-Algebra Examples

$\frac{64 x^{3} - 1}{2 x^{2} - 6 x + 18} \div \frac{48 x^{2} + 12 x + 3}{6 x^{3} + 162}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{64 x^{3} - 1}{2 x^{2} - 6 x + 18} \cdot \frac{6 x^{3} + 162}{48 x^{2} + 12 x + 3}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $64 x^{3}$ as ${(4 x)}^{3}$ .

$\frac{{(4 x)}^{3} - 1}{2 x^{2} - 6 x + 18} \cdot \frac{6 x^{3} + 162}{48 x^{2} + 12 x + 3}$

Step 2.2

Rewrite $1$ as $1^{3}$ .

$\frac{{(4 x)}^{3} - 1^{3}}{2 x^{2} - 6 x + 18} \cdot \frac{6 x^{3} + 162}{48 x^{2} + 12 x + 3}$

Step 2.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 4 x$ and $b = 1$ .

$\frac{(4 x - 1) ({(4 x)}^{2} + 4 x \cdot 1 + 1^{2})}{2 x^{2} - 6 x + 18} \cdot \frac{6 x^{3} + 162}{48 x^{2} + 12 x + 3}$

Step 2.4

Simplify.

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Step 2.4.1

Apply the product rule to $4 x$ .

$\frac{(4 x - 1) (4^{2} x^{2} + 4 x \cdot 1 + 1^{2})}{2 x^{2} - 6 x + 18} \cdot \frac{6 x^{3} + 162}{48 x^{2} + 12 x + 3}$

Step 2.4.2

Raise $4$ to the power of $2$ .

$\frac{(4 x - 1) (16 x^{2} + 4 x \cdot 1 + 1^{2})}{2 x^{2} - 6 x + 18} \cdot \frac{6 x^{3} + 162}{48 x^{2} + 12 x + 3}$

Step 2.4.3

Multiply $4$ by $1$ .

$\frac{(4 x - 1) (16 x^{2} + 4 x + 1^{2})}{2 x^{2} - 6 x + 18} \cdot \frac{6 x^{3} + 162}{48 x^{2} + 12 x + 3}$

Step 2.4.4

One to any power is one.

$\frac{(4 x - 1) (16 x^{2} + 4 x + 1)}{2 x^{2} - 6 x + 18} \cdot \frac{6 x^{3} + 162}{48 x^{2} + 12 x + 3}$

Step 3

Factor $2$ out of $2 x^{2} - 6 x + 18$ .

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Step 3.1

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

$\frac{(4 x - 1) (16 x^{2} + 4 x + 1)}{2 (x^{2}) - 6 x + 18} \cdot \frac{6 x^{3} + 162}{48 x^{2} + 12 x + 3}$

Step 3.2

Factor $2$ out of $- 6 x$ .

$\frac{(4 x - 1) (16 x^{2} + 4 x + 1)}{2 (x^{2}) + 2 (- 3 x) + 18} \cdot \frac{6 x^{3} + 162}{48 x^{2} + 12 x + 3}$

Step 3.3

Factor $2$ out of $18$ .

$\frac{(4 x - 1) (16 x^{2} + 4 x + 1)}{2 x^{2} + 2 (- 3 x) + 2 \cdot 9} \cdot \frac{6 x^{3} + 162}{48 x^{2} + 12 x + 3}$

Step 3.4

Factor $2$ out of $2 x^{2} + 2 (- 3 x)$ .

$\frac{(4 x - 1) (16 x^{2} + 4 x + 1)}{2 (x^{2} - 3 x) + 2 \cdot 9} \cdot \frac{6 x^{3} + 162}{48 x^{2} + 12 x + 3}$

Step 3.5

Factor $2$ out of $2 (x^{2} - 3 x) + 2 \cdot 9$ .

$\frac{(4 x - 1) (16 x^{2} + 4 x + 1)}{2 (x^{2} - 3 x + 9)} \cdot \frac{6 x^{3} + 162}{48 x^{2} + 12 x + 3}$

Step 4

Cancel the common factor of $6 x^{3} + 162$ and $48 x^{2} + 12 x + 3$ .

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Step 4.1

Factor $3$ out of $6 x^{3}$ .

$\frac{(4 x - 1) (16 x^{2} + 4 x + 1)}{2 (x^{2} - 3 x + 9)} \cdot \frac{3 (2 x^{3}) + 162}{48 x^{2} + 12 x + 3}$

Step 4.2

Factor $3$ out of $162$ .

$\frac{(4 x - 1) (16 x^{2} + 4 x + 1)}{2 (x^{2} - 3 x + 9)} \cdot \frac{3 (2 x^{3}) + 3 \cdot 54}{48 x^{2} + 12 x + 3}$

Step 4.3

Factor $3$ out of $3 (2 x^{3}) + 3 (54)$ .

$\frac{(4 x - 1) (16 x^{2} + 4 x + 1)}{2 (x^{2} - 3 x + 9)} \cdot \frac{3 (2 x^{3} + 54)}{48 x^{2} + 12 x + 3}$

Step 4.4

Cancel the common factors.

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Step 4.4.1

Factor $3$ out of $48 x^{2}$ .

$\frac{(4 x - 1) (16 x^{2} + 4 x + 1)}{2 (x^{2} - 3 x + 9)} \cdot \frac{3 (2 x^{3} + 54)}{3 (16 x^{2}) + 12 x + 3}$

Step 4.4.2

Factor $3$ out of $12 x$ .

$\frac{(4 x - 1) (16 x^{2} + 4 x + 1)}{2 (x^{2} - 3 x + 9)} \cdot \frac{3 (2 x^{3} + 54)}{3 (16 x^{2}) + 3 (4 x) + 3}$

Step 4.4.3

Factor $3$ out of $3 (16 x^{2}) + 3 (4 x)$ .

$\frac{(4 x - 1) (16 x^{2} + 4 x + 1)}{2 (x^{2} - 3 x + 9)} \cdot \frac{3 (2 x^{3} + 54)}{3 (16 x^{2} + 4 x) + 3}$

Step 4.4.4

Factor $3$ out of $3$ .

$\frac{(4 x - 1) (16 x^{2} + 4 x + 1)}{2 (x^{2} - 3 x + 9)} \cdot \frac{3 (2 x^{3} + 54)}{3 (16 x^{2} + 4 x) + 3 (1)}$

Step 4.4.5

Factor $3$ out of $3 (16 x^{2} + 4 x) + 3 (1)$ .

$\frac{(4 x - 1) (16 x^{2} + 4 x + 1)}{2 (x^{2} - 3 x + 9)} \cdot \frac{3 (2 x^{3} + 54)}{3 (16 x^{2} + 4 x + 1)}$

Step 4.4.6

Cancel the common factor.

$\frac{(4 x - 1) (16 x^{2} + 4 x + 1)}{2 (x^{2} - 3 x + 9)} \cdot \frac{3 (2 x^{3} + 54)}{3 (16 x^{2} + 4 x + 1)}$

Step 4.4.7

Rewrite the expression.

$\frac{(4 x - 1) (16 x^{2} + 4 x + 1)}{2 (x^{2} - 3 x + 9)} \cdot \frac{2 x^{3} + 54}{16 x^{2} + 4 x + 1}$

Step 5

Cancel the common factor of $16 x^{2} + 4 x + 1$ .

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Step 5.1

Factor $16 x^{2} + 4 x + 1$ out of $(4 x - 1) (16 x^{2} + 4 x + 1)$ .

$\frac{(16 x^{2} + 4 x + 1) (4 x - 1)}{2 (x^{2} - 3 x + 9)} \cdot \frac{2 x^{3} + 54}{16 x^{2} + 4 x + 1}$

Step 5.2

Cancel the common factor.

$\frac{(16 x^{2} + 4 x + 1) (4 x - 1)}{2 (x^{2} - 3 x + 9)} \cdot \frac{2 x^{3} + 54}{16 x^{2} + 4 x + 1}$

Step 5.3

Rewrite the expression.

$\frac{4 x - 1}{2 (x^{2} - 3 x + 9)} (2 x^{3} + 54)$

Step 6

Multiply $\frac{4 x - 1}{2 (x^{2} - 3 x + 9)}$ by $2 x^{3} + 54$ .

$\frac{(4 x - 1) (2 x^{3} + 54)}{2 (x^{2} - 3 x + 9)}$

Step 7

Cancel the common factor of $2 x^{3} + 54$ and $2$ .

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Step 7.1

Factor $2$ out of $(4 x - 1) (2 x^{3} + 54)$ .

$\frac{2 ((4 x - 1) (x^{3} + 27))}{2 (x^{2} - 3 x + 9)}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Cancel the common factor.

$\frac{2 ((4 x - 1) (x^{3} + 27))}{2 (x^{2} - 3 x + 9)}$

Step 7.2.2

Rewrite the expression.

$\frac{(4 x - 1) (x^{3} + 27)}{x^{2} - 3 x + 9}$

Step 8

Simplify the numerator.

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Step 8.1

Rewrite $27$ as $3^{3}$ .

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

Step 8.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 3$ .

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

Step 8.3

Simplify.

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Step 8.3.1

Multiply $3$ by $- 1$ .

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

Step 8.3.2

Raise $3$ to the power of $2$ .

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

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

Step 9

Cancel the common factor of $x^{2} - 3 x + 9$ .

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Step 9.1

Cancel the common factor.

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

Step 9.2

Divide $(4 x - 1) (x + 3)$ by $1$ .

$(4 x - 1) (x + 3)$

Step 10

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

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Step 10.1

Apply the distributive property.

$4 x (x + 3) - 1 (x + 3)$

Step 10.2

Apply the distributive property.

$4 x \cdot x + 4 x \cdot 3 - 1 (x + 3)$

Step 10.3

Apply the distributive property.

$4 x \cdot x + 4 x \cdot 3 - 1 x - 1 \cdot 3$

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

Multiply $x$ by $x$ by adding the exponents.

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Step 11.1.1.1

Move $x$ .

$4 (x \cdot x) + 4 x \cdot 3 - 1 x - 1 \cdot 3$

Step 11.1.1.2

Multiply $x$ by $x$ .

$4 x^{2} + 4 x \cdot 3 - 1 x - 1 \cdot 3$

Step 11.1.2

Multiply $3$ by $4$ .

$4 x^{2} + 12 x - 1 x - 1 \cdot 3$

Step 11.1.3

Rewrite $- 1 x$ as $- x$ .

$4 x^{2} + 12 x - x - 1 \cdot 3$

Step 11.1.4

Multiply $- 1$ by $3$ .

$4 x^{2} + 12 x - x - 3$

Step 11.2

Subtract $x$ from $12 x$ .

$4 x^{2} + 11 x - 3$