Solve for x 2-3(x+1)^(2/3)=-25

$2 - 3 {(x + 1)}^{\frac{2}{3}} = - 25$

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

Subtract $2$ from both sides of the equation.

$- 3 {(x + 1)}^{\frac{2}{3}} = - 25 - 2$

Subtract $2$ from $- 25$ .

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

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

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

Simplify the left side.

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

Apply the product rule to $- 3 {(x + 1)}^{\frac{2}{3}}$ .

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Simplify the expression.

Multiply ${(- 3)}^{\frac{3}{2}}$ by $1$ .

${(- 3)}^{\frac{3}{2}} x + {(- 3)}^{\frac{3}{2}} = \pm {(- 27)}^{\frac{3}{2}}$

Reorder factors in ${(- 3)}^{\frac{3}{2}} x + {(- 3)}^{\frac{3}{2}}$ .

$x {(- 3)}^{\frac{3}{2}} + {(- 3)}^{\frac{3}{2}} = \pm {(- 27)}^{\frac{3}{2}}$

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

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

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

Subtract ${(- 3)}^{\frac{3}{2}}$ from both sides of the equation.

$x {(- 3)}^{\frac{3}{2}} = {(- 27)}^{\frac{3}{2}} - {(- 3)}^{\frac{3}{2}}$

Divide each term in $x {(- 3)}^{\frac{3}{2}} = {(- 27)}^{\frac{3}{2}} - {(- 3)}^{\frac{3}{2}}$ by ${(- 3)}^{\frac{3}{2}}$ and simplify.

Divide each term in $x {(- 3)}^{\frac{3}{2}} = {(- 27)}^{\frac{3}{2}} - {(- 3)}^{\frac{3}{2}}$ by ${(- 3)}^{\frac{3}{2}}$ .

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

Simplify the left side.

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Simplify each term.

Use the power of quotient rule $\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ .

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

Divide $- 27$ by $- 3$ .

$x = 9^{\frac{3}{2}} + \frac{- {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Rewrite $9$ as $3^{2}$ .

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Cancel the common factor.

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

Divide $- 1$ by $1$ .

$x = 27 - 1$

Subtract $1$ from $27$ .

$x = 26$

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

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

Subtract ${(- 3)}^{\frac{3}{2}}$ from both sides of the equation.

$x {(- 3)}^{\frac{3}{2}} = - {(- 27)}^{\frac{3}{2}} - {(- 3)}^{\frac{3}{2}}$

Divide each term in $x {(- 3)}^{\frac{3}{2}} = - {(- 27)}^{\frac{3}{2}} - {(- 3)}^{\frac{3}{2}}$ by ${(- 3)}^{\frac{3}{2}}$ and simplify.

Divide each term in $x {(- 3)}^{\frac{3}{2}} = - {(- 27)}^{\frac{3}{2}} - {(- 3)}^{\frac{3}{2}}$ by ${(- 3)}^{\frac{3}{2}}$ .

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

Simplify the left side.

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

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

Use the power of quotient rule $\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ .

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

Divide $- 27$ by $- 3$ .

$x = - 9^{\frac{3}{2}} + \frac{- {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Rewrite $9$ as $3^{2}$ .

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Multiply $- 1$ by $27$ .

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

Cancel the common factor.

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

Divide $- 1$ by $1$ .

$x = - 27 - 1$

Subtract $1$ from $- 27$ .

$x = - 28$

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

$x = 26, - 28$

Solve for x 2-3(x+1)^(2/3)=-25

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