Algebra Examples

$\sqrt[3]{x + 5} = x - 1$

Step 1

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

${\sqrt[3]{x + 5}}^{3} = {(x - 1)}^{3}$

Step 2

Simplify each side of the equation.

Tap for more steps...

Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x + 5}$ as ${(x + 5)}^{\frac{1}{3}}$ .

${({(x + 5)}^{\frac{1}{3}})}^{3} = {(x - 1)}^{3}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify ${({(x + 5)}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 2.2.1.1

Multiply the exponents in ${({(x + 5)}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 2.2.1.1.1

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

${(x + 5)}^{\frac{1}{3} \cdot 3} = {(x - 1)}^{3}$

Step 2.2.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.1.1.2.1

Cancel the common factor.

${(x + 5)}^{\frac{1}{3} \cdot 3} = {(x - 1)}^{3}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x + 5)}^{1} = {(x - 1)}^{3}$

Step 2.2.1.2

Simplify.

$x + 5 = {(x - 1)}^{3}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

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

Tap for more steps...

Step 2.3.1.1

Use the Binomial Theorem.

$x + 5 = x^{3} + 3 x^{2} \cdot - 1 + 3 x {(- 1)}^{2} + {(- 1)}^{3}$

Step 2.3.1.2

Simplify each term.

Tap for more steps...

Step 2.3.1.2.1

Multiply $- 1$ by $3$ .

$x + 5 = x^{3} - 3 x^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}$

Step 2.3.1.2.2

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

$x + 5 = x^{3} - 3 x^{2} + 3 x \cdot 1 + {(- 1)}^{3}$

Step 2.3.1.2.3

Multiply $3$ by $1$ .

$x + 5 = x^{3} - 3 x^{2} + 3 x + {(- 1)}^{3}$

Step 2.3.1.2.4

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

$x + 5 = x^{3} - 3 x^{2} + 3 x - 1$

Step 3

Solve for $x$ .

Tap for more steps...

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.

$x^{3} - 3 x^{2} + 3 x - 1 = x + 5$

Step 3.2

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

Tap for more steps...

Step 3.2.1

Subtract $x$ from both sides of the equation.

$x^{3} - 3 x^{2} + 3 x - 1 - x = 5$

Step 3.2.2

Subtract $x$ from $3 x$ .

$x^{3} - 3 x^{2} + 2 x - 1 = 5$

Step 3.3

Subtract $5$ from both sides of the equation.

$x^{3} - 3 x^{2} + 2 x - 1 - 5 = 0$

Step 3.4

Subtract $5$ from $- 1$ .

$x^{3} - 3 x^{2} + 2 x - 6 = 0$

Step 3.5

Factor the left side of the equation.

Tap for more steps...

Step 3.5.1

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.5.1.1

Group the first two terms and the last two terms.

$(x^{3} - 3 x^{2}) + 2 x - 6 = 0$

Step 3.5.1.2

Factor out the greatest common factor (GCF) from each group.

$x^{2} (x - 3) + 2 (x - 3) = 0$

Step 3.5.2

Factor the polynomial by factoring out the greatest common factor, $x - 3$ .

$(x - 3) (x^{2} + 2) = 0$

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 $0$ .

$x - 3 = 0$

$x^{2} + 2 = 0$

Step 3.7

Set $x - 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.7.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 3.7.2

Add $3$ to both sides of the equation.

$x = 3$

Step 3.8

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

Tap for more steps...

Step 3.8.1

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

$x^{2} + 2 = 0$

Step 3.8.2

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

Tap for more steps...

Step 3.8.2.1

Subtract $2$ from both sides of the equation.

$x^{2} = - 2$

Step 3.8.2.2

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

$x = \pm \sqrt{- 2}$

Step 3.8.2.3

Simplify $\pm \sqrt{- 2}$ .

Tap for more steps...

Step 3.8.2.3.1

Rewrite $- 2$ as $- 1 (2)$ .

$x = \pm \sqrt{- 1 (2)}$

Step 3.8.2.3.2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

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

Step 3.8.2.3.3

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

$x = \pm i \sqrt{2}$

Step 3.8.2.4

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 3.8.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i \sqrt{2}$

Step 3.8.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i \sqrt{2}$

Step 3.8.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i \sqrt{2}, - i \sqrt{2}$

Step 3.9

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

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