Algebra Examples

$\sqrt[3]{8 x^{3} + 27} = 2 x + 3$

Step 1

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

${\sqrt[3]{8 x^{3} + 27}}^{3} = {(2 x + 3)}^{3}$

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 $\sqrt[3]{8 x^{3} + 27}$ as ${(8 x^{3} + 27)}^{\frac{1}{3}}$ .

${({(8 x^{3} + 27)}^{\frac{1}{3}})}^{3} = {(2 x + 3)}^{3}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify ${({(8 x^{3} + 27)}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 2.2.1.1

Multiply the exponents in ${({(8 x^{3} + 27)}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 2.2.1.1.1

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

${(8 x^{3} + 27)}^{\frac{1}{3} \cdot 3} = {(2 x + 3)}^{3}$

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.

${(8 x^{3} + 27)}^{\frac{1}{3} \cdot 3} = {(2 x + 3)}^{3}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(8 x^{3} + 27)}^{1} = {(2 x + 3)}^{3}$

Step 2.2.1.2

Simplify.

$8 x^{3} + 27 = {(2 x + 3)}^{3}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Simplify ${(2 x + 3)}^{3}$ .

Tap for more steps...

Step 2.3.1.1

Use the Binomial Theorem.

$8 x^{3} + 27 = {(2 x)}^{3} + 3 {(2 x)}^{2} \cdot 3 + 3 (2 x) \cdot 3^{2} + 3^{3}$

Step 2.3.1.2

Simplify each term.

Tap for more steps...

Step 2.3.1.2.1

Apply the product rule to $2 x$ .

$8 x^{3} + 27 = 2^{3} x^{3} + 3 {(2 x)}^{2} \cdot 3 + 3 (2 x) \cdot 3^{2} + 3^{3}$

Step 2.3.1.2.2

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

$8 x^{3} + 27 = 8 x^{3} + 3 {(2 x)}^{2} \cdot 3 + 3 (2 x) \cdot 3^{2} + 3^{3}$

Step 2.3.1.2.3

Apply the product rule to $2 x$ .

$8 x^{3} + 27 = 8 x^{3} + 3 (2^{2} x^{2}) \cdot 3 + 3 (2 x) \cdot 3^{2} + 3^{3}$

Step 2.3.1.2.4

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

$8 x^{3} + 27 = 8 x^{3} + 3 (4 x^{2}) \cdot 3 + 3 (2 x) \cdot 3^{2} + 3^{3}$

Step 2.3.1.2.5

Multiply $4$ by $3$ .

$8 x^{3} + 27 = 8 x^{3} + 12 x^{2} \cdot 3 + 3 (2 x) \cdot 3^{2} + 3^{3}$

Step 2.3.1.2.6

Multiply $3$ by $12$ .

$8 x^{3} + 27 = 8 x^{3} + 36 x^{2} + 3 (2 x) \cdot 3^{2} + 3^{3}$

Step 2.3.1.2.7

Multiply $3$ by $3^{2}$ by adding the exponents.

Tap for more steps...

Step 2.3.1.2.7.1

Move $3^{2}$ .

$8 x^{3} + 27 = 8 x^{3} + 36 x^{2} + 3^{2} \cdot 3 (2 x) + 3^{3}$

Step 2.3.1.2.7.2

Multiply $3^{2}$ by $3$ .

Tap for more steps...

Step 2.3.1.2.7.2.1

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

$8 x^{3} + 27 = 8 x^{3} + 36 x^{2} + 3^{2} \cdot 3^{1} (2 x) + 3^{3}$

Step 2.3.1.2.7.2.2

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

$8 x^{3} + 27 = 8 x^{3} + 36 x^{2} + 3^{2 + 1} (2 x) + 3^{3}$

Step 2.3.1.2.7.3

Add $2$ and $1$ .

$8 x^{3} + 27 = 8 x^{3} + 36 x^{2} + 3^{3} (2 x) + 3^{3}$

Step 2.3.1.2.8

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

$8 x^{3} + 27 = 8 x^{3} + 36 x^{2} + 27 (2 x) + 3^{3}$

Step 2.3.1.2.9

Multiply $2$ by $27$ .

$8 x^{3} + 27 = 8 x^{3} + 36 x^{2} + 54 x + 3^{3}$

Step 2.3.1.2.10

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

$8 x^{3} + 27 = 8 x^{3} + 36 x^{2} + 54 x + 27$

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.

$8 x^{3} + 36 x^{2} + 54 x + 27 = 8 x^{3} + 27$

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

$8 x^{3} + 36 x^{2} + 54 x + 27 - 8 x^{3} = 27$

Step 3.2.2

Combine the opposite terms in $8 x^{3} + 36 x^{2} + 54 x + 27 - 8 x^{3}$ .

Tap for more steps...

Step 3.2.2.1

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

$36 x^{2} + 54 x + 27 + 0 = 27$

Step 3.2.2.2

Add $36 x^{2} + 54 x + 27$ and $0$ .

$36 x^{2} + 54 x + 27 = 27$

Step 3.3

Subtract $27$ from both sides of the equation.

$36 x^{2} + 54 x + 27 - 27 = 0$

Step 3.4

Combine the opposite terms in $36 x^{2} + 54 x + 27 - 27$ .

Tap for more steps...

Step 3.4.1

Subtract $27$ from $27$ .

$36 x^{2} + 54 x + 0 = 0$

Step 3.4.2

Add $36 x^{2} + 54 x$ and $0$ .

$36 x^{2} + 54 x = 0$

Step 3.5

Factor $18 x$ out of $36 x^{2} + 54 x$ .

Tap for more steps...

Step 3.5.1

Factor $18 x$ out of $36 x^{2}$ .

$18 x (2 x) + 54 x = 0$

Step 3.5.2

Factor $18 x$ out of $54 x$ .

$18 x (2 x) + 18 x (3) = 0$

Step 3.5.3

Factor $18 x$ out of $18 x (2 x) + 18 x (3)$ .

$18 x (2 x + 3) = 0$

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 $0$ .

$x = 0$

$2 x + 3 = 0$

Step 3.7

Set $x$ equal to $0$ .

$x = 0$

Step 3.8

Set $2 x + 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.8.1

Set $2 x + 3$ equal to $0$ .

$2 x + 3 = 0$

Step 3.8.2

Solve $2 x + 3 = 0$ for $x$ .

Tap for more steps...

Step 3.8.2.1

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 3.8.2.2

Divide each term in $2 x = - 3$ by $2$ and simplify.

Tap for more steps...

Step 3.8.2.2.1

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 3.8.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.8.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.8.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 3.8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Step 3.8.2.2.3

Simplify the right side.

Tap for more steps...

Step 3.8.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 3.9

The final solution is all the values that make $18 x (2 x + 3) = 0$ true.

$x = 0, - \frac{3}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 0, - \frac{3}{2}$

Decimal Form:

$x = 0, - 1.5$

Mixed Number Form:

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