Algebra Examples

$\sqrt[3]{8 x^{3} - 1} = 2 x - 1$

Step 1

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

${\sqrt[3]{8 x^{3} - 1}}^{3} = {(2 x - 1)}^{3}$

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 $\sqrt[3]{8 x^{3} - 1}$ as ${(8 x^{3} - 1)}^{\frac{1}{3}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(8 x^{3} - 1)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(8 x^{3} - 1)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1.1

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

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

${(8 x^{3} - 1)}^{1} = {(2 x - 1)}^{3}$

Step 2.2.1.2

Simplify.

$8 x^{3} - 1 = {(2 x - 1)}^{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Use the Binomial Theorem.

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

Step 2.3.1.2

Simplify each term.

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Step 2.3.1.2.1

Apply the product rule to $2 x$ .

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

Step 2.3.1.2.2

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

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

Step 2.3.1.2.3

Apply the product rule to $2 x$ .

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

Step 2.3.1.2.4

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

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

Step 2.3.1.2.5

Multiply $4$ by $3$ .

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

Step 2.3.1.2.6

Multiply $- 1$ by $12$ .

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

Step 2.3.1.2.7

Multiply $2$ by $3$ .

$8 x^{3} - 1 = 8 x^{3} - 12 x^{2} + 6 x {(- 1)}^{2} + {(- 1)}^{3}$

Step 2.3.1.2.8

Raise $- 1$ to the power of $2$ .

$8 x^{3} - 1 = 8 x^{3} - 12 x^{2} + 6 x \cdot 1 + {(- 1)}^{3}$

Step 2.3.1.2.9

Multiply $6$ by $1$ .

$8 x^{3} - 1 = 8 x^{3} - 12 x^{2} + 6 x + {(- 1)}^{3}$

Step 2.3.1.2.10

Raise $- 1$ to the power of $3$ .

$8 x^{3} - 1 = 8 x^{3} - 12 x^{2} + 6 x - 1$

Step 3

Solve for $x$ .

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Step 3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$8 x^{3} - 12 x^{2} + 6 x - 1 = 8 x^{3} - 1$

Step 3.2

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

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Step 3.2.1

Subtract $8 x^{3}$ from both sides of the equation.

$8 x^{3} - 12 x^{2} + 6 x - 1 - 8 x^{3} = - 1$

Step 3.2.2

Combine the opposite terms in $8 x^{3} - 12 x^{2} + 6 x - 1 - 8 x^{3}$ .

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Step 3.2.2.1

Subtract $8 x^{3}$ from $8 x^{3}$ .

$- 12 x^{2} + 6 x - 1 + 0 = - 1$

Step 3.2.2.2

Add $- 12 x^{2} + 6 x - 1$ and $0$ .

$- 12 x^{2} + 6 x - 1 = - 1$

Step 3.3

Add $1$ to both sides of the equation.

$- 12 x^{2} + 6 x - 1 + 1 = 0$

Step 3.4

Combine the opposite terms in $- 12 x^{2} + 6 x - 1 + 1$ .

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Step 3.4.1

Add $- 1$ and $1$ .

$- 12 x^{2} + 6 x + 0 = 0$

Step 3.4.2

Add $- 12 x^{2} + 6 x$ and $0$ .

$- 12 x^{2} + 6 x = 0$

Step 3.5

Factor $- 6 x$ out of $- 12 x^{2} + 6 x$ .

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Step 3.5.1

Factor $- 6 x$ out of $- 12 x^{2}$ .

$- 6 x (2 x) + 6 x = 0$

Step 3.5.2

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

$- 6 x (2 x) - 6 x \cdot - 1 = 0$

Step 3.5.3

Factor $- 6 x$ out of $- 6 x (2 x) - 6 x (- 1)$ .

$- 6 x (2 x - 1) = 0$

Step 3.6

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

$x = 0$

$2 x - 1 = 0$

Step 3.7

Set $x$ equal to $0$ .

$x = 0$

Step 3.8

Set $2 x - 1$ equal to $0$ and solve for $x$ .

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Step 3.8.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 3.8.2

Solve $2 x - 1 = 0$ for $x$ .

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Step 3.8.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 3.8.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 3.8.2.2.1

Divide each term in $2 x = 1$ by $2$ .

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

Step 3.8.2.2.2

Simplify the left side.

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Step 3.8.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.8.2.2.2.1.1

Cancel the common factor.

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

Step 3.8.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.9

The final solution is all the values that make $- 6 x (2 x - 1) = 0$ true.

$x = 0, \frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{1}{2}$

Decimal Form:

$x = 0, 0.5$