Solve for n (3n+2)^(1/3)+1=0

${(3 n + 2)}^{\frac{1}{3}} + 1 = 0$

Step 1

Subtract $1$ from both sides of the equation.

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

Step 2

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

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

Step 3

Simplify the exponent.

Simplify the left side.

Simplify ${({(3 n + 2)}^{\frac{1}{3}})}^{3}$ .

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

$3 n + 2 = - 1$

Step 4

Solve for $n$ .

Move all terms not containing $n$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$3 n = - 1 - 2$

Subtract $2$ from $- 1$ .

$3 n = - 3$

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

Divide each term in $3 n = - 3$ by $3$ .

$\frac{3 n}{3} = \frac{- 3}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 n}{3} = \frac{- 3}{3}$

Divide $n$ by $1$ .

$n = \frac{- 3}{3}$

Simplify the right side.

Divide $- 3$ by $3$ .

$n = - 1$