Find All Complex Solutions 4(3-x)^(4/3)-5=59

$4 {(3 - x)}^{\frac{4}{3}} - 5 = 59$

Subtract $59$ from both sides of the equation.

$4 {(3 - x)}^{\frac{4}{3}} - 5 - 59 = 0$

Subtract $59$ from $- 5$ .

$4 {(3 - x)}^{\frac{4}{3}} - 64 = 0$

Factor the left side of the equation.

Factor $4$ out of $4 {(3 - x)}^{\frac{4}{3}} - 64$ .

Factor $4$ out of $4 {(3 - x)}^{\frac{4}{3}}$ .

$4 ({(3 - x)}^{\frac{4}{3}}) - 64 = 0$

Factor $4$ out of $- 64$ .

$4 {(3 - x)}^{\frac{4}{3}} + 4 \cdot - 16 = 0$

Factor $4$ out of $4 {(3 - x)}^{\frac{4}{3}} + 4 \cdot - 16$ .

$4 ({(3 - x)}^{\frac{4}{3}} - 16) = 0$

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

$4 ({({(3 - x)}^{\frac{2}{3}})}^{2} - 16) = 0$

Rewrite $16$ as $4^{2}$ .

$4 ({({(3 - x)}^{\frac{2}{3}})}^{2} - 4^{2}) = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(3 - x)}^{\frac{2}{3}}$ and $b = 4$ .

$4 (({(3 - x)}^{\frac{2}{3}} + 4) ({(3 - x)}^{\frac{2}{3}} - 4)) = 0$

Factor.

Simplify.

Rewrite ${(3 - x)}^{\frac{2}{3}}$ as ${({(3 - x)}^{\frac{1}{3}})}^{2}$ .

$4 (({(3 - x)}^{\frac{2}{3}} + 4) ({({(3 - x)}^{\frac{1}{3}})}^{2} - 4)) = 0$

Rewrite $4$ as $2^{2}$ .

$4 (({(3 - x)}^{\frac{2}{3}} + 4) ({({(3 - x)}^{\frac{1}{3}})}^{2} - 2^{2})) = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(3 - x)}^{\frac{1}{3}}$ and $b = 2$ .

$4 (({(3 - x)}^{\frac{2}{3}} + 4) (({(3 - x)}^{\frac{1}{3}} + 2) ({(3 - x)}^{\frac{1}{3}} - 2))) = 0$

Remove unnecessary parentheses.

$4 (({(3 - x)}^{\frac{2}{3}} + 4) ({(3 - x)}^{\frac{1}{3}} + 2) ({(3 - x)}^{\frac{1}{3}} - 2)) = 0$

Remove unnecessary parentheses.

$4 ({(3 - x)}^{\frac{2}{3}} + 4) ({(3 - x)}^{\frac{1}{3}} + 2) ({(3 - x)}^{\frac{1}{3}} - 2) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(3 - x)}^{\frac{2}{3}} + 4 = 0$

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

${(3 - x)}^{\frac{1}{3}} - 2 = 0$

Set ${(3 - x)}^{\frac{2}{3}} + 4$ equal to $0$ and solve for $x$ .

Set ${(3 - x)}^{\frac{2}{3}} + 4$ equal to $0$ .

${(3 - x)}^{\frac{2}{3}} + 4 = 0$

Solve ${(3 - x)}^{\frac{2}{3}} + 4 = 0$ for $x$ .

Subtract $4$ from both sides of the equation.

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

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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.

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

Subtract $3$ from both sides of the equation.

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

Divide each term in $- x = {(- 4)}^{\frac{3}{2}} - 3$ by $- 1$ and simplify.

Divide each term in $- x = {(- 4)}^{\frac{3}{2}} - 3$ by $- 1$ .

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{{(- 4)}^{\frac{3}{2}}}{- 1}$ .

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

Rewrite $- 1 \cdot {(- 4)}^{\frac{3}{2}}$ as $- {(- 4)}^{\frac{3}{2}}$ .

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

Divide $- 3$ by $- 1$ .

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

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

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

Subtract $3$ from both sides of the equation.

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

Divide each term in $- x = - {(- 4)}^{\frac{3}{2}} - 3$ by $- 1$ and simplify.

Divide each term in $- x = - {(- 4)}^{\frac{3}{2}} - 3$ by $- 1$ .

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Simplify each term.

Dividing two negative values results in a positive value.

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

Divide ${(- 4)}^{\frac{3}{2}}$ by $1$ .

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

Divide $- 3$ by $- 1$ .

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

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

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

Set ${(3 - x)}^{\frac{1}{3}} + 2$ equal to $0$ and solve for $x$ .

Set ${(3 - x)}^{\frac{1}{3}} + 2$ equal to $0$ .

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

Solve ${(3 - x)}^{\frac{1}{3}} + 2 = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

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

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

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

$3 - x = - 8$

Solve for $x$ .

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

Subtract $3$ from both sides of the equation.

$- x = - 8 - 3$

Subtract $3$ from $- 8$ .

$- x = - 11$

Divide each term in $- x = - 11$ by $- 1$ and simplify.

Divide each term in $- x = - 11$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 11}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 11}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 11}{- 1}$

Simplify the right side.

Divide $- 11$ by $- 1$ .

$x = 11$

Set ${(3 - x)}^{\frac{1}{3}} - 2$ equal to $0$ and solve for $x$ .

Set ${(3 - x)}^{\frac{1}{3}} - 2$ equal to $0$ .

${(3 - x)}^{\frac{1}{3}} - 2 = 0$

Solve ${(3 - x)}^{\frac{1}{3}} - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

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

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

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$3 - x = 2^{3}$

Simplify the right side.

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

$3 - x = 8$

Solve for $x$ .

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

Subtract $3$ from both sides of the equation.

$- x = 8 - 3$

Subtract $3$ from $8$ .

$- x = 5$

Divide each term in $- x = 5$ by $- 1$ and simplify.

Divide each term in $- x = 5$ by $- 1$ .

$\frac{- x}{- 1} = \frac{5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{5}{- 1}$

Divide $x$ by $1$ .

$x = \frac{5}{- 1}$

Simplify the right side.

Divide $5$ by $- 1$ .

$x = - 5$

The final solution is all the values that make $4 ({(3 - x)}^{\frac{2}{3}} + 4) ({(3 - x)}^{\frac{1}{3}} + 2) ({(3 - x)}^{\frac{1}{3}} - 2) = 0$ true.

$x = - {(- 4)}^{\frac{3}{2}} + 3, {(- 4)}^{\frac{3}{2}} + 3, 11, - 5$

Find All Complex Solutions 4(3-x)^(4/3)-5=59

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