Solve for x 3+(4-x)^(2/3)=12

$3 + {(4 - x)}^{\frac{2}{3}} = 12$

Step 1

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

Subtract $3$ from both sides of the equation.

${(4 - x)}^{\frac{2}{3}} = 12 - 3$

Subtract $3$ from $12$ .

${(4 - x)}^{\frac{2}{3}} = 9$

Step 2

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

${({(4 - x)}^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 9^{\frac{3}{2}}$

Step 3

Simplify the exponent.

Simplify the left side.

Simplify ${({(4 - x)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Multiply the exponents in ${({(4 - x)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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

${(4 - x)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 9^{\frac{3}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 - x)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 9^{\frac{3}{2}}$

Rewrite the expression.

${(4 - x)}^{\frac{1}{3} \cdot 3} = \pm 9^{\frac{3}{2}}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(4 - x)}^{\frac{1}{3} \cdot 3} = \pm 9^{\frac{3}{2}}$

Rewrite the expression.

${(4 - x)}^{1} = \pm 9^{\frac{3}{2}}$

Simplify.

$4 - x = \pm 9^{\frac{3}{2}}$

Simplify the right side.

Simplify $\pm 9^{\frac{3}{2}}$ .

Simplify the expression.

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

$4 - x = \pm {(3^{2})}^{\frac{3}{2}}$

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

$4 - x = \pm 3^{2 (\frac{3}{2})}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 - x = \pm 3^{2 (\frac{3}{2})}$

Rewrite the expression.

$4 - x = \pm 3^{3}$

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

$4 - x = \pm 27$

Step 4

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.

$4 - x = 27$

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

Subtract $4$ from both sides of the equation.

$- x = 27 - 4$

Subtract $4$ from $27$ .

$- x = 23$

Divide each term in $- x = 23$ by $- 1$ and simplify.

Divide each term in $- x = 23$ by $- 1$ .

$\frac{- x}{- 1} = \frac{23}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{23}{- 1}$

Divide $x$ by $1$ .

$x = \frac{23}{- 1}$

Simplify the right side.

Divide $23$ by $- 1$ .

$x = - 23$

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

$4 - x = - 27$

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

Subtract $4$ from both sides of the equation.

$- x = - 27 - 4$

Subtract $4$ from $- 27$ .

$- x = - 31$

Divide each term in $- x = - 31$ by $- 1$ and simplify.

Divide each term in $- x = - 31$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 31}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 31}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 31}{- 1}$

Simplify the right side.

Divide $- 31$ by $- 1$ .

$x = 31$

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

$x = - 23, 31$

Solve for x 3+(4-x)^(2/3)=12

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