Precalculus Examples

${(12 x - 1)}^{\frac{1}{3}} (2) (x^{2} - 1) (4 x) + {(x^{2} - 1)}^{2} (\frac{1}{4}) {(12 x - 1)}^{- \frac{2}{3}} (32)$

Step 1

Simplify each term.

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Step 1.1

Move $2$ to the left of ${(12 x - 1)}^{\frac{1}{3}}$ .

$2 \cdot {(12 x - 1)}^{\frac{1}{3}} (x^{2} - 1) (4 x) + {(x^{2} - 1)}^{2} (\frac{1}{4}) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.2

Apply the distributive property.

$(2 {(12 x - 1)}^{\frac{1}{3}} x^{2} + 2 {(12 x - 1)}^{\frac{1}{3}} \cdot - 1) (4 x) + {(x^{2} - 1)}^{2} (\frac{1}{4}) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.3

Multiply $- 1$ by $2$ .

$(2 {(12 x - 1)}^{\frac{1}{3}} x^{2} - 2 {(12 x - 1)}^{\frac{1}{3}}) (4 x) + {(x^{2} - 1)}^{2} (\frac{1}{4}) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.4

Apply the distributive property.

$2 {(12 x - 1)}^{\frac{1}{3}} x^{2} (4 x) - 2 {(12 x - 1)}^{\frac{1}{3}} (4 x) + {(x^{2} - 1)}^{2} (\frac{1}{4}) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.5

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

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Step 1.5.1

Move $x$ .

$2 {(12 x - 1)}^{\frac{1}{3}} (x \cdot x^{2}) \cdot 4 - 2 {(12 x - 1)}^{\frac{1}{3}} (4 x) + {(x^{2} - 1)}^{2} (\frac{1}{4}) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.5.2

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

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Step 1.5.2.1

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

$2 {(12 x - 1)}^{\frac{1}{3}} (x^{1} x^{2}) \cdot 4 - 2 {(12 x - 1)}^{\frac{1}{3}} (4 x) + {(x^{2} - 1)}^{2} (\frac{1}{4}) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.5.2.2

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

$2 {(12 x - 1)}^{\frac{1}{3}} x^{1 + 2} \cdot 4 - 2 {(12 x - 1)}^{\frac{1}{3}} (4 x) + {(x^{2} - 1)}^{2} (\frac{1}{4}) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.5.3

Add $1$ and $2$ .

$2 {(12 x - 1)}^{\frac{1}{3}} x^{3} \cdot 4 - 2 {(12 x - 1)}^{\frac{1}{3}} (4 x) + {(x^{2} - 1)}^{2} (\frac{1}{4}) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.6

Rewrite using the commutative property of multiplication.

$2 {(12 x - 1)}^{\frac{1}{3}} x^{3} \cdot 4 - 2 \cdot 4 {(12 x - 1)}^{\frac{1}{3}} x + {(x^{2} - 1)}^{2} (\frac{1}{4}) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.7

Simplify each term.

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Step 1.7.1

Multiply $4$ by $2$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 2 \cdot 4 {(12 x - 1)}^{\frac{1}{3}} x + {(x^{2} - 1)}^{2} (\frac{1}{4}) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.7.2

Multiply $- 2$ by $4$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + {(x^{2} - 1)}^{2} (\frac{1}{4}) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.8

Rewrite ${(x^{2} - 1)}^{2}$ as $(x^{2} - 1) (x^{2} - 1)$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (x^{2} - 1) (x^{2} - 1) \frac{1}{4} \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.9

Expand $(x^{2} - 1) (x^{2} - 1)$ using the FOIL Method.

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Step 1.9.1

Apply the distributive property.

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (x^{2} (x^{2} - 1) - 1 (x^{2} - 1)) \frac{1}{4} \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.9.2

Apply the distributive property.

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (x^{2} x^{2} + x^{2} \cdot - 1 - 1 (x^{2} - 1)) \frac{1}{4} \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.9.3

Apply the distributive property.

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (x^{2} x^{2} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1) \frac{1}{4} \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.10

Simplify and combine like terms.

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Step 1.10.1

Simplify each term.

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Step 1.10.1.1

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

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Step 1.10.1.1.1

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

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (x^{2 + 2} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1) \frac{1}{4} \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.10.1.1.2

Add $2$ and $2$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (x^{4} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1) \frac{1}{4} \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.10.1.2

Move $- 1$ to the left of $x^{2}$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (x^{4} - 1 \cdot x^{2} - 1 x^{2} - 1 \cdot - 1) \frac{1}{4} \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.10.1.3

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (x^{4} - x^{2} - 1 x^{2} - 1 \cdot - 1) \frac{1}{4} \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.10.1.4

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (x^{4} - x^{2} - x^{2} - 1 \cdot - 1) \frac{1}{4} \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.10.1.5

Multiply $- 1$ by $- 1$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (x^{4} - x^{2} - x^{2} + 1) \frac{1}{4} \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.10.2

Subtract $x^{2}$ from $- x^{2}$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (x^{4} - 2 x^{2} + 1) \frac{1}{4} \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.11

Apply the distributive property.

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (x^{4} \frac{1}{4} - 2 x^{2} \frac{1}{4} + 1 (\frac{1}{4})) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.12

Simplify.

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Step 1.12.1

Combine $x^{4}$ and $\frac{1}{4}$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (\frac{x^{4}}{4} - 2 x^{2} \frac{1}{4} + 1 (\frac{1}{4})) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.12.2

Cancel the common factor of $2$ .

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Step 1.12.2.1

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

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (\frac{x^{4}}{4} + 2 (- x^{2}) \frac{1}{4} + 1 (\frac{1}{4})) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.12.2.2

Factor $2$ out of $4$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (\frac{x^{4}}{4} + 2 (- x^{2}) \frac{1}{2 (2)} + 1 (\frac{1}{4})) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.12.2.3

Cancel the common factor.

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (\frac{x^{4}}{4} + 2 (- x^{2}) \frac{1}{2 \cdot 2} + 1 (\frac{1}{4})) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.12.2.4

Rewrite the expression.

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (\frac{x^{4}}{4} - x^{2} \frac{1}{2} + 1 (\frac{1}{4})) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.12.3

Combine $\frac{1}{2}$ and $x^{2}$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (\frac{x^{4}}{4} - \frac{x^{2}}{2} + 1 (\frac{1}{4})) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.12.4

Multiply $\frac{1}{4}$ by $1$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (\frac{x^{4}}{4} - \frac{x^{2}}{2} + \frac{1}{4}) \cdot ({(12 x - 1)}^{- \frac{2}{3}} (32))$

Step 1.13

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (\frac{x^{4}}{4} - \frac{x^{2}}{2} + \frac{1}{4}) \cdot (\frac{1}{{(12 x - 1)}^{\frac{2}{3}}} \cdot 32)$

Step 1.14

Combine $\frac{1}{{(12 x - 1)}^{\frac{2}{3}}}$ and $32$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + (\frac{x^{4}}{4} - \frac{x^{2}}{2} + \frac{1}{4}) \cdot \frac{32}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.15

Multiply $\frac{x^{4}}{4} - \frac{x^{2}}{2} + \frac{1}{4}$ by $\frac{32}{{(12 x - 1)}^{\frac{2}{3}}}$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{(\frac{x^{4}}{4} - \frac{x^{2}}{2} + \frac{1}{4}) \cdot 32}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.16

Simplify the numerator.

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Step 1.16.1

Factor using the perfect square rule.

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Step 1.16.1.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{(\frac{{(x^{2})}^{2}}{4} - \frac{x^{2}}{2} + \frac{1}{4}) \cdot 32}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.16.1.2

Rewrite $4$ as $2^{2}$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{(\frac{{(x^{2})}^{2}}{2^{2}} - \frac{x^{2}}{2} + \frac{1}{4}) \cdot 32}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.16.1.3

Rewrite $\frac{{(x^{2})}^{2}}{2^{2}}$ as ${(\frac{x^{2}}{2})}^{2}$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{({(\frac{x^{2}}{2})}^{2} - \frac{x^{2}}{2} + \frac{1}{4}) \cdot 32}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.16.1.4

Rewrite $1$ as $1^{2}$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{({(\frac{x^{2}}{2})}^{2} - \frac{x^{2}}{2} + \frac{1^{2}}{4}) \cdot 32}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.16.1.5

Rewrite $4$ as $2^{2}$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{({(\frac{x^{2}}{2})}^{2} - \frac{x^{2}}{2} + \frac{1^{2}}{2^{2}}) \cdot 32}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.16.1.6

Rewrite $\frac{1^{2}}{2^{2}}$ as ${(\frac{1}{2})}^{2}$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{({(\frac{x^{2}}{2})}^{2} - \frac{x^{2}}{2} + {(\frac{1}{2})}^{2}) \cdot 32}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.16.1.7

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$\frac{x^{2}}{2} = 2 \cdot \frac{x^{2}}{2} \cdot \frac{1}{2}$

Step 1.16.1.8

Rewrite the polynomial.

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{({(\frac{x^{2}}{2})}^{2} - 2 \cdot \frac{x^{2}}{2} \cdot \frac{1}{2} + {(\frac{1}{2})}^{2}) \cdot 32}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.16.1.9

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = \frac{x^{2}}{2}$ and $b = \frac{1}{2}$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{{(\frac{x^{2}}{2} - \frac{1}{2})}^{2} \cdot 32}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.16.2

Combine the numerators over the common denominator.

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{{(\frac{x^{2} - 1}{2})}^{2} \cdot 32}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.16.3

Simplify the numerator.

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Step 1.16.3.1

Rewrite $1$ as $1^{2}$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{{(\frac{x^{2} - 1^{2}}{2})}^{2} \cdot 32}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.16.3.2

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

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{{(\frac{(x + 1) (x - 1)}{2})}^{2} \cdot 32}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.16.4

Apply the product rule to $\frac{(x + 1) (x - 1)}{2}$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{\frac{{((x + 1) (x - 1))}^{2}}{2^{2}} \cdot 32}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.16.5

Apply the product rule to $(x + 1) (x - 1)$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{\frac{{(x + 1)}^{2} {(x - 1)}^{2}}{2^{2}} \cdot 32}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.16.6

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

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{\frac{{(x + 1)}^{2} {(x - 1)}^{2}}{4} \cdot 32}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.17

Combine $\frac{{(x + 1)}^{2} {(x - 1)}^{2}}{4}$ and $32$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{\frac{{(x + 1)}^{2} {(x - 1)}^{2} \cdot 32}{4}}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.18

Reduce the expression by cancelling the common factors.

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Step 1.18.1

Reduce the expression $\frac{{(x + 1)}^{2} {(x - 1)}^{2} \cdot 32}{4}$ by cancelling the common factors.

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Step 1.18.1.1

Factor $4$ out of ${(x + 1)}^{2} {(x - 1)}^{2} \cdot 32$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{\frac{4 ({(x + 1)}^{2} {(x - 1)}^{2} \cdot 8)}{4}}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.18.1.2

Factor $4$ out of $4$ .

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{\frac{4 ({(x + 1)}^{2} {(x - 1)}^{2} \cdot 8)}{4 (1)}}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.18.1.3

Cancel the common factor.

$8 {(12 x - 1)}^{\frac{1}{3}} x^{3} - 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{\frac{4 ({(x + 1)}^{2} {(x - 1)}^{2} \cdot 8)}{4 \cdot 1}}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 1.18.1.4

Rewrite the expression.

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

Step 1.18.2

Divide ${(x + 1)}^{2} {(x - 1)}^{2} \cdot 8$ by $1$ .

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

Step 1.19

Move $8$ to the left of ${(x + 1)}^{2} {(x - 1)}^{2}$ .

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

Step 2

To write $8 {(12 x - 1)}^{\frac{1}{3}} x^{3}$ as a fraction with a common denominator, multiply by $\frac{{(12 x - 1)}^{\frac{2}{3}}}{{(12 x - 1)}^{\frac{2}{3}}}$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Factor $8$ out of $8 {(12 x - 1)}^{\frac{1}{3}} x^{3} {(12 x - 1)}^{\frac{2}{3}} + 8 {(x + 1)}^{2} {(x - 1)}^{2}$ .

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Step 4.1.1

Factor $8$ out of $8 {(12 x - 1)}^{\frac{1}{3}} x^{3} {(12 x - 1)}^{\frac{2}{3}}$ .

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

Step 4.1.2

Factor $8$ out of $8 {(x + 1)}^{2} {(x - 1)}^{2}$ .

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

Step 4.1.3

Factor $8$ out of $8 ({(12 x - 1)}^{\frac{1}{3}} x^{3} {(12 x - 1)}^{\frac{2}{3}}) + 8 ({(x + 1)}^{2} {(x - 1)}^{2})$ .

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

Step 4.2

Multiply ${(12 x - 1)}^{\frac{1}{3}}$ by ${(12 x - 1)}^{\frac{2}{3}}$ by adding the exponents.

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Step 4.2.1

Move ${(12 x - 1)}^{\frac{2}{3}}$ .

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

Step 4.2.2

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

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

Step 4.2.3

Combine the numerators over the common denominator.

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

Step 4.2.4

Add $2$ and $1$ .

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

Step 4.2.5

Divide $3$ by $3$ .

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

Step 4.3

Simplify ${(12 x - 1)}^{1} x^{3}$ .

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 4.5.1

Move $x^{3}$ .

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

Step 4.5.2

Multiply $x^{3}$ by $x$ .

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Step 4.5.2.1

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

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

Step 4.5.2.2

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

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

Step 4.5.3

Add $3$ and $1$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - 1 x^{3} + {(x + 1)}^{2} {(x - 1)}^{2})}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.6

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

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + {(x + 1)}^{2} {(x - 1)}^{2})}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.7

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x + 1) (x + 1) {(x - 1)}^{2})}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.8

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

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Step 4.8.1

Apply the distributive property.

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x (x + 1) + 1 (x + 1)) {(x - 1)}^{2})}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.8.2

Apply the distributive property.

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x \cdot x + x \cdot 1 + 1 (x + 1)) {(x - 1)}^{2})}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.8.3

Apply the distributive property.

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) {(x - 1)}^{2})}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.9

Simplify and combine like terms.

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Step 4.9.1

Simplify each term.

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Step 4.9.1.1

Multiply $x$ by $x$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) {(x - 1)}^{2})}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.9.1.2

Multiply $x$ by $1$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x^{2} + x + 1 x + 1 \cdot 1) {(x - 1)}^{2})}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.9.1.3

Multiply $x$ by $1$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x^{2} + x + x + 1 \cdot 1) {(x - 1)}^{2})}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.9.1.4

Multiply $1$ by $1$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x^{2} + x + x + 1) {(x - 1)}^{2})}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.9.2

Add $x$ and $x$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x^{2} + 2 x + 1) {(x - 1)}^{2})}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.10

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x^{2} + 2 x + 1) ((x - 1) (x - 1)))}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.11

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

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Step 4.11.1

Apply the distributive property.

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x^{2} + 2 x + 1) (x (x - 1) - 1 (x - 1)))}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.11.2

Apply the distributive property.

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x^{2} + 2 x + 1) (x \cdot x + x \cdot - 1 - 1 (x - 1)))}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.11.3

Apply the distributive property.

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x^{2} + 2 x + 1) (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1))}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.12

Simplify and combine like terms.

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Step 4.12.1

Simplify each term.

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Step 4.12.1.1

Multiply $x$ by $x$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x^{2} + 2 x + 1) (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1))}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.12.1.2

Move $- 1$ to the left of $x$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x^{2} + 2 x + 1) (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1))}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.12.1.3

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

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x^{2} + 2 x + 1) (x^{2} - x - 1 x - 1 \cdot - 1))}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.12.1.4

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

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x^{2} + 2 x + 1) (x^{2} - x - x - 1 \cdot - 1))}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.12.1.5

Multiply $- 1$ by $- 1$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x^{2} + 2 x + 1) (x^{2} - x - x + 1))}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.12.2

Subtract $x$ from $- x$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + (x^{2} + 2 x + 1) (x^{2} - 2 x + 1))}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.13

Expand $(x^{2} + 2 x + 1) (x^{2} - 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{2} x^{2} + x^{2} (- 2 x) + x^{2} \cdot 1 + 2 x \cdot x^{2} + 2 x (- 2 x) + 2 x \cdot 1 + 1 x^{2} + 1 (- 2 x) + 1 \cdot 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.14

Combine the opposite terms in $x^{2} x^{2} + x^{2} (- 2 x) + x^{2} \cdot 1 + 2 x \cdot x^{2} + 2 x (- 2 x) + 2 x \cdot 1 + 1 x^{2} + 1 (- 2 x) + 1 \cdot 1$ .

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Step 4.14.1

Reorder the factors in the terms $x^{2} (- 2 x)$ and $2 x \cdot x^{2}$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{2} x^{2} - 2 x^{2} x + x^{2} \cdot 1 + 2 x^{2} x + 2 x (- 2 x) + 2 x \cdot 1 + 1 x^{2} + 1 (- 2 x) + 1 \cdot 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.14.2

Add $- 2 x^{2} x$ and $2 x^{2} x$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{2} x^{2} + x^{2} \cdot 1 + 0 + 2 x (- 2 x) + 2 x \cdot 1 + 1 x^{2} + 1 (- 2 x) + 1 \cdot 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.14.3

Add $x^{2} x^{2} + x^{2} \cdot 1$ and $0$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{2} x^{2} + x^{2} \cdot 1 + 2 x (- 2 x) + 2 x \cdot 1 + 1 x^{2} + 1 (- 2 x) + 1 \cdot 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.15

Simplify each term.

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Step 4.15.1

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

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Step 4.15.1.1

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

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{2 + 2} + x^{2} \cdot 1 + 2 x (- 2 x) + 2 x \cdot 1 + 1 x^{2} + 1 (- 2 x) + 1 \cdot 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.15.1.2

Add $2$ and $2$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{4} + x^{2} \cdot 1 + 2 x (- 2 x) + 2 x \cdot 1 + 1 x^{2} + 1 (- 2 x) + 1 \cdot 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.15.2

Multiply $x^{2}$ by $1$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{4} + x^{2} + 2 x (- 2 x) + 2 x \cdot 1 + 1 x^{2} + 1 (- 2 x) + 1 \cdot 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.15.3

Rewrite using the commutative property of multiplication.

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{4} + x^{2} + 2 \cdot - 2 x \cdot x + 2 x \cdot 1 + 1 x^{2} + 1 (- 2 x) + 1 \cdot 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.15.4

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

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Step 4.15.4.1

Move $x$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{4} + x^{2} + 2 \cdot - 2 (x \cdot x) + 2 x \cdot 1 + 1 x^{2} + 1 (- 2 x) + 1 \cdot 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.15.4.2

Multiply $x$ by $x$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{4} + x^{2} + 2 \cdot - 2 x^{2} + 2 x \cdot 1 + 1 x^{2} + 1 (- 2 x) + 1 \cdot 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.15.5

Multiply $2$ by $- 2$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{4} + x^{2} - 4 x^{2} + 2 x \cdot 1 + 1 x^{2} + 1 (- 2 x) + 1 \cdot 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.15.6

Multiply $2$ by $1$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{4} + x^{2} - 4 x^{2} + 2 x + 1 x^{2} + 1 (- 2 x) + 1 \cdot 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.15.7

Multiply $x^{2}$ by $1$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{4} + x^{2} - 4 x^{2} + 2 x + x^{2} + 1 (- 2 x) + 1 \cdot 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.15.8

Multiply $- 2 x$ by $1$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{4} + x^{2} - 4 x^{2} + 2 x + x^{2} - 2 x + 1 \cdot 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.15.9

Multiply $1$ by $1$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{4} + x^{2} - 4 x^{2} + 2 x + x^{2} - 2 x + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.16

Combine the opposite terms in $x^{4} + x^{2} - 4 x^{2} + 2 x + x^{2} - 2 x + 1$ .

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Step 4.16.1

Subtract $2 x$ from $2 x$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{4} + x^{2} - 4 x^{2} + x^{2} + 0 + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.16.2

Add $x^{4} + x^{2} - 4 x^{2} + x^{2}$ and $0$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{4} + x^{2} - 4 x^{2} + x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.17

Subtract $4 x^{2}$ from $x^{2}$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{4} - 3 x^{2} + x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.18

Add $- 3 x^{2}$ and $x^{2}$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (12 x^{4} - x^{3} + x^{4} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 4.19

Add $12 x^{4}$ and $x^{4}$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x + \frac{8 (13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 5

To write $- 8 {(12 x - 1)}^{\frac{1}{3}} x$ as a fraction with a common denominator, multiply by $\frac{{(12 x - 1)}^{\frac{2}{3}}}{{(12 x - 1)}^{\frac{2}{3}}}$ .

$- 8 {(12 x - 1)}^{\frac{1}{3}} x \cdot \frac{{(12 x - 1)}^{\frac{2}{3}}}{{(12 x - 1)}^{\frac{2}{3}}} + \frac{8 (13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 6

Simplify terms.

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Step 6.1

Combine $- 8 {(12 x - 1)}^{\frac{1}{3}} x$ and $\frac{{(12 x - 1)}^{\frac{2}{3}}}{{(12 x - 1)}^{\frac{2}{3}}}$ .

$\frac{- 8 {(12 x - 1)}^{\frac{1}{3}} x {(12 x - 1)}^{\frac{2}{3}}}{{(12 x - 1)}^{\frac{2}{3}}} + \frac{8 (13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 6.2

Combine the numerators over the common denominator.

$\frac{- 8 {(12 x - 1)}^{\frac{1}{3}} x {(12 x - 1)}^{\frac{2}{3}} + 8 (13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 7

Simplify the numerator.

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Step 7.1

Factor $8$ out of $- 8 {(12 x - 1)}^{\frac{1}{3}} x {(12 x - 1)}^{\frac{2}{3}} + 8 (13 x^{4} - x^{3} - 2 x^{2} + 1)$ .

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Step 7.1.1

Factor $8$ out of $- 8 {(12 x - 1)}^{\frac{1}{3}} x {(12 x - 1)}^{\frac{2}{3}}$ .

$\frac{8 (- {(12 x - 1)}^{\frac{1}{3}} x {(12 x - 1)}^{\frac{2}{3}}) + 8 (13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 7.1.2

Factor $8$ out of $8 (- {(12 x - 1)}^{\frac{1}{3}} x {(12 x - 1)}^{\frac{2}{3}}) + 8 (13 x^{4} - x^{3} - 2 x^{2} + 1)$ .

$\frac{8 (- {(12 x - 1)}^{\frac{1}{3}} x {(12 x - 1)}^{\frac{2}{3}} + 13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 7.2

Multiply ${(12 x - 1)}^{\frac{1}{3}}$ by ${(12 x - 1)}^{\frac{2}{3}}$ by adding the exponents.

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Step 7.2.1

Move ${(12 x - 1)}^{\frac{2}{3}}$ .

$\frac{8 (- ({(12 x - 1)}^{\frac{2}{3}} {(12 x - 1)}^{\frac{1}{3}}) x + 13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 7.2.2

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

$\frac{8 (- {(12 x - 1)}^{\frac{2}{3} + \frac{1}{3}} x + 13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 7.2.3

Combine the numerators over the common denominator.

$\frac{8 (- {(12 x - 1)}^{\frac{2 + 1}{3}} x + 13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 7.2.4

Add $2$ and $1$ .

$\frac{8 (- {(12 x - 1)}^{\frac{3}{3}} x + 13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 7.2.5

Divide $3$ by $3$ .

$\frac{8 (- {(12 x - 1)}^{1} x + 13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 7.3

Simplify $- {(12 x - 1)}^{1} x$ .

$\frac{8 (- (12 x - 1) x + 13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 7.4

Apply the distributive property.

$\frac{8 ((- (12 x) - - 1) x + 13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 7.5

Multiply $12$ by $- 1$ .

$\frac{8 ((- 12 x - - 1) x + 13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 7.6

Multiply $- 1$ by $- 1$ .

$\frac{8 ((- 12 x + 1) x + 13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 7.7

Apply the distributive property.

$\frac{8 (- 12 x \cdot x + 1 x + 13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 7.8

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

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Step 7.8.1

Move $x$ .

$\frac{8 (- 12 (x \cdot x) + 1 x + 13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 7.8.2

Multiply $x$ by $x$ .

$\frac{8 (- 12 x^{2} + 1 x + 13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 7.9

Multiply $x$ by $1$ .

$\frac{8 (- 12 x^{2} + x + 13 x^{4} - x^{3} - 2 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 7.10

Subtract $2 x^{2}$ from $- 12 x^{2}$ .

$\frac{8 (x + 13 x^{4} - x^{3} - 14 x^{2} + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$

Step 7.11

Reorder terms.

$\frac{8 (13 x^{4} - x^{3} - 14 x^{2} + x + 1)}{{(12 x - 1)}^{\frac{2}{3}}}$