Finite Math Examples

$\frac{4 x \sqrt{x^{3} - 1} - (\frac{3 x^{4}}{\sqrt{x^{3} - 1}})}{x^{3} - 1} = 0$

Step 1

Set the numerator equal to zero.

$4 x \sqrt{x^{3} - 1} - \frac{3 x^{4}}{\sqrt{x^{3} - 1}} = 0$

Step 2

Solve the equation for $x$ .

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Step 2.1

Simplify both sides of the equation.

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

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

$4 x \sqrt{x^{3} - 1^{3}} - \frac{3 x^{4}}{\sqrt{x^{3} - 1}} = 0$

Step 2.1.1.2

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 $a = x$ and $b = 1$ .

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

Step 2.1.1.3

Simplify.

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Step 2.1.1.3.1

Multiply $x$ by $1$ .

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

Step 2.1.1.3.2

One to any power is one.

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

Step 2.1.1.4

Simplify the denominator.

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Step 2.1.1.4.1

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

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

Step 2.1.1.4.2

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 $a = x$ and $b = 1$ .

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

Step 2.1.1.4.3

Simplify.

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Step 2.1.1.4.3.1

Multiply $x$ by $1$ .

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

Step 2.1.1.4.3.2

One to any power is one.

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

Step 2.1.1.5

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

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

Step 2.1.1.6

Combine and simplify the denominator.

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Step 2.1.1.6.1

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

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

Step 2.1.1.6.2

Raise $\sqrt{(x - 1) (x^{2} + x + 1)}$ to the power of $1$ .

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

Step 2.1.1.6.3

Raise $\sqrt{(x - 1) (x^{2} + x + 1)}$ to the power of $1$ .

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

Step 2.1.1.6.4

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

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

Step 2.1.1.6.5

Add $1$ and $1$ .

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

Step 2.1.1.6.6

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

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Step 2.1.1.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(x - 1) (x^{2} + x + 1)}$ as ${((x - 1) (x^{2} + x + 1))}^{\frac{1}{2}}$ .

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

Step 2.1.1.6.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.1.1.6.6.3

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

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

Step 2.1.1.6.6.4

Cancel the common factor of $2$ .

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Step 2.1.1.6.6.4.1

Cancel the common factor.

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

Step 2.1.1.6.6.4.2

Rewrite the expression.

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

Step 2.1.1.6.6.5

Simplify.

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

Step 2.1.2

To write $4 x \sqrt{(x - 1) (x^{2} + x + 1)}$ as a fraction with a common denominator, multiply by $\frac{(x - 1) (x^{2} + x + 1)}{(x - 1) (x^{2} + x + 1)}$ .

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

Step 2.1.3

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

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

Step 2.1.4

Combine the numerators over the common denominator.

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

Step 2.1.5

Simplify the numerator.

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Step 2.1.5.1

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

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Step 2.1.5.1.1

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

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

Step 2.1.5.1.2

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

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

Step 2.1.5.1.3

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

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

Step 2.1.5.2

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

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

Step 2.1.5.3

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

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Step 2.1.5.3.1

Reorder the factors in the terms $x \cdot 1$ and $- 1 x$ .

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

Step 2.1.5.3.2

Subtract $1 x$ from $1 x$ .

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

Step 2.1.5.3.3

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

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

Step 2.1.5.4

Simplify each term.

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Step 2.1.5.4.1

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

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Step 2.1.5.4.1.1

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

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Step 2.1.5.4.1.1.1

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

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

Step 2.1.5.4.1.1.2

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

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

Step 2.1.5.4.1.2

Add $1$ and $2$ .

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

Step 2.1.5.4.2

Multiply $x$ by $x$ .

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

Step 2.1.5.4.3

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

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

Step 2.1.5.4.4

Multiply $- 1$ by $1$ .

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

Step 2.1.5.5

Combine the opposite terms in $x^{3} + x^{2} - x^{2} - 1$ .

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Step 2.1.5.5.1

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

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

Step 2.1.5.5.2

Add $x^{3}$ and $0$ .

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

Step 2.1.5.6

Apply the distributive property.

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

Step 2.1.5.7

Multiply $4$ by $- 1$ .

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

Step 2.1.5.8

Subtract $3 x^{3}$ from $4 x^{3}$ .

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

Step 2.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(x - 1) (x^{2} + x + 1)}$ as ${((x - 1) (x^{2} + x + 1))}^{\frac{1}{2}}$ .

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

Step 2.3

Simplify the numerator.

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Step 2.3.1

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

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

Step 2.3.2

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

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Step 2.3.2.1

Reorder the factors in the terms $x \cdot 1$ and $- 1 x$ .

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

Step 2.3.2.2

Subtract $1 x$ from $1 x$ .

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

Step 2.3.2.3

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

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

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

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Step 2.3.3.1.1

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

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Step 2.3.3.1.1.1

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

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

Step 2.3.3.1.1.2

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

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

Step 2.3.3.1.2

Add $1$ and $2$ .

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

Step 2.3.3.2

Multiply $x$ by $x$ .

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

Step 2.3.3.3

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

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

Step 2.3.3.4

Multiply $- 1$ by $1$ .

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

Step 2.3.4

Combine the opposite terms in $x^{3} + x^{2} - x^{2} - 1$ .

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Step 2.3.4.1

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

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

Step 2.3.4.2

Add $x^{3}$ and $0$ .

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

Step 2.4

Find the LCD of the terms in the equation.

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Step 2.4.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$(x - 1) (x^{2} + x + 1), 1$

Step 2.4.2

The LCM of one and any expression is the expression.

$(x - 1) (x^{2} + x + 1)$

Step 2.5

Multiply each term in $\frac{x {(x^{3} - 1)}^{\frac{1}{2}} (x^{3} - 4)}{(x - 1) (x^{2} + x + 1)} = 0$ by $(x - 1) (x^{2} + x + 1)$ to eliminate the fractions.

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Step 2.5.1

Multiply each term in $\frac{x {(x^{3} - 1)}^{\frac{1}{2}} (x^{3} - 4)}{(x - 1) (x^{2} + x + 1)} = 0$ by $(x - 1) (x^{2} + x + 1)$ .

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

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $(x - 1) (x^{2} + x + 1)$ .

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Step 2.5.2.1.1

Cancel the common factor.

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

Step 2.5.2.1.2

Rewrite the expression.

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

Step 2.5.2.2

Apply the distributive property.

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

Step 2.5.2.3

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

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Step 2.5.2.3.1

Move $x^{3}$ .

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

Step 2.5.2.3.2

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

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Step 2.5.2.3.2.1

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

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

Step 2.5.2.3.2.2

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

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

Step 2.5.2.3.3

Add $3$ and $1$ .

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

Step 2.5.2.4

Move $- 4$ to the left of $x {(x^{3} - 1)}^{\frac{1}{2}}$ .

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

Step 2.5.2.5

Remove parentheses.

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

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

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

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

Step 2.5.3.2

Simplify terms.

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Step 2.5.3.2.1

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

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Step 2.5.3.2.1.1

Reorder the factors in the terms $x \cdot 1$ and $- 1 x$ .

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

Step 2.5.3.2.1.2

Subtract $1 x$ from $1 x$ .

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

Step 2.5.3.2.1.3

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

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

Step 2.5.3.2.2

Simplify each term.

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Step 2.5.3.2.2.1

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

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Step 2.5.3.2.2.1.1

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

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Step 2.5.3.2.2.1.1.1

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

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

Step 2.5.3.2.2.1.1.2

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

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

Step 2.5.3.2.2.1.2

Add $1$ and $2$ .

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

Step 2.5.3.2.2.2

Multiply $x$ by $x$ .

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

Step 2.5.3.2.2.3

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

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

Step 2.5.3.2.2.4

Multiply $- 1$ by $1$ .

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

Step 2.5.3.2.3

Simplify by adding terms.

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Step 2.5.3.2.3.1

Combine the opposite terms in $x^{3} + x^{2} - x^{2} - 1$ .

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Step 2.5.3.2.3.1.1

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

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

Step 2.5.3.2.3.1.2

Add $x^{3}$ and $0$ .

$x^{4} {(x^{3} - 1)}^{\frac{1}{2}} - 4 x {(x^{3} - 1)}^{\frac{1}{2}} = 0 (x^{3} - 1)$

Step 2.5.3.2.3.2

Multiply $0$ by $x^{3} - 1$ .

$x^{4} {(x^{3} - 1)}^{\frac{1}{2}} - 4 x {(x^{3} - 1)}^{\frac{1}{2}} = 0$

Step 2.6

Solve the equation.

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Step 2.6.1

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

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Step 2.6.1.1

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

$x {(x^{3} - 1)}^{\frac{1}{2}} (x^{3}) - 4 x {(x^{3} - 1)}^{\frac{1}{2}} = 0$

Step 2.6.1.2

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

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

Step 2.6.1.3

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

$x {(x^{3} - 1)}^{\frac{1}{2}} (x^{3} - 4) = 0$

Step 2.6.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

${(x^{3} - 1)}^{\frac{1}{2}} = 0$

$x^{3} - 4 = 0$

Step 2.6.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.6.4

Set ${(x^{3} - 1)}^{\frac{1}{2}}$ equal to $0$ and solve for $x$ .

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Step 2.6.4.1

Set ${(x^{3} - 1)}^{\frac{1}{2}}$ equal to $0$ .

${(x^{3} - 1)}^{\frac{1}{2}} = 0$

Step 2.6.4.2

Solve ${(x^{3} - 1)}^{\frac{1}{2}} = 0$ for $x$ .

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Step 2.6.4.2.1

Set the $x^{3} - 1$ equal to $0$ .

$x^{3} - 1 = 0$

Step 2.6.4.2.2

Solve for $x$ .

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Step 2.6.4.2.2.1

Add $1$ to both sides of the equation.

$x^{3} = 1$

Step 2.6.4.2.2.2

Subtract $1$ from both sides of the equation.

$x^{3} - 1 = 0$

Step 2.6.4.2.2.3

Factor the left side of the equation.

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Step 2.6.4.2.2.3.1

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

$x^{3} - 1^{3} = 0$

Step 2.6.4.2.2.3.2

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 $a = x$ and $b = 1$ .

$(x - 1) (x^{2} + x \cdot 1 + 1^{2}) = 0$

Step 2.6.4.2.2.3.3

Simplify.

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Step 2.6.4.2.2.3.3.1

Multiply $x$ by $1$ .

$(x - 1) (x^{2} + x + 1^{2}) = 0$

Step 2.6.4.2.2.3.3.2

One to any power is one.

$(x - 1) (x^{2} + x + 1) = 0$

Step 2.6.4.2.2.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

$x^{2} + x + 1 = 0$

Step 2.6.4.2.2.5

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 2.6.4.2.2.5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.6.4.2.2.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.6.4.2.2.6

Set $x^{2} + x + 1$ equal to $0$ and solve for $x$ .

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Step 2.6.4.2.2.6.1

Set $x^{2} + x + 1$ equal to $0$ .

$x^{2} + x + 1 = 0$

Step 2.6.4.2.2.6.2

Solve $x^{2} + x + 1 = 0$ for $x$ .

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Step 2.6.4.2.2.6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.6.4.2.2.6.2.2

Substitute the values $a = 1$ , $b = 1$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 2.6.4.2.2.6.2.3

Simplify.

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Step 2.6.4.2.2.6.2.3.1

Simplify the numerator.

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Step 2.6.4.2.2.6.2.3.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.6.4.2.2.6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 2.6.4.2.2.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.6.4.2.2.6.2.3.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 2.6.4.2.2.6.2.3.1.3

Subtract $4$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 2.6.4.2.2.6.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 2.6.4.2.2.6.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 2.6.4.2.2.6.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 2.6.4.2.2.6.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2}$

Step 2.6.4.2.2.6.2.4

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 2.6.4.2.2.7

The final solution is all the values that make $(x - 1) (x^{2} + x + 1) = 0$ true.

$x = 1, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 2.6.5

Set $x^{3} - 4$ equal to $0$ and solve for $x$ .

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Step 2.6.5.1

Set $x^{3} - 4$ equal to $0$ .

$x^{3} - 4 = 0$

Step 2.6.5.2

Solve $x^{3} - 4 = 0$ for $x$ .

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Step 2.6.5.2.1

Add $4$ to both sides of the equation.

$x^{3} = 4$

Step 2.6.5.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{4}$

Step 2.6.6

The final solution is all the values that make $x {(x^{3} - 1)}^{\frac{1}{2}} (x^{3} - 4) = 0$ true.

$x = 0, 1, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}, \sqrt[3]{4}$