Algebra Examples

$\sqrt[3]{2 x^{3} - 1} = x$

Step 1

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{2 x^{3} - 1}}^{3} = x^{3}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 x^{3} - 1}$ as ${(2 x^{3} - 1)}^{\frac{1}{3}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

$2 x^{3} - 1 = x^{3}$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.1.1

Subtract $x^{3}$ from both sides of the equation.

$2 x^{3} - 1 - x^{3} = 0$

Step 3.1.2

Subtract $x^{3}$ from $2 x^{3}$ .

$x^{3} - 1 = 0$

Step 3.2

Add $1$ to both sides of the equation.

$x^{3} = 1$

Step 3.3

Subtract $1$ from both sides of the equation.

$x^{3} - 1 = 0$

Step 3.4

Factor the left side of the equation.

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Step 3.4.1

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

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

Step 3.4.2

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 $b = 1$ .

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

Step 3.4.3

Simplify.

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Step 3.4.3.1

Multiply $x$ by $1$ .

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

Step 3.4.3.2

One to any power is one.

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

Step 3.5

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 3.6

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

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Step 3.6.1

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

$x - 1 = 0$

Step 3.6.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3.7

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

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Step 3.7.1

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

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

Step 3.7.2

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

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Step 3.7.2.1

Use the quadratic formula to find the solutions.

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

Step 3.7.2.2

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 3.7.2.3

Simplify.

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Step 3.7.2.3.1

Simplify the numerator.

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Step 3.7.2.3.1.1

One to any power is one.

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

Step 3.7.2.3.1.2

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

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Step 3.7.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 3.7.2.3.1.2.2

Multiply $- 4$ by $1$ .

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

Step 3.7.2.3.1.3

Subtract $4$ from $1$ .

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

Step 3.7.2.3.1.4

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

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

Step 3.7.2.3.1.5

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

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

Step 3.7.2.3.1.6

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

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

Step 3.7.2.3.2

Multiply $2$ by $1$ .

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

Step 3.7.2.4

The final answer is the combination of both solutions.

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

Step 3.8

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

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