Algebra Examples

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

Step 1

Subtract $59$ from both sides of the equation.

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

Step 2

Subtract $59$ from $- 5$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Factor $4$ out of $- 64$ .

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

Step 3.3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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 $b = 4$ .

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

Step 7

Factor.

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Step 7.1

Simplify.

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Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.1.3

Factor.

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Step 7.1.3.1

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 $b = 2$ .

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

Step 7.1.3.2

Remove unnecessary parentheses.

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

Step 7.2

Remove unnecessary parentheses.

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

Step 8

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

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

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

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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Step 9.2.1

Subtract $4$ from both sides of the equation.

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

Step 9.2.2

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

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

Step 9.2.3

Simplify the left side.

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Step 9.2.3.1

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

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Step 9.2.3.1.1

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

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Step 9.2.3.1.1.1

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

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

Step 9.2.3.1.1.2

Cancel the common factor of $2$ .

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Step 9.2.3.1.1.2.1

Cancel the common factor.

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

Step 9.2.3.1.1.2.2

Rewrite the expression.

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

Step 9.2.3.1.1.3

Cancel the common factor of $3$ .

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Step 9.2.3.1.1.3.1

Cancel the common factor.

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

Step 9.2.3.1.1.3.2

Rewrite the expression.

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

Step 9.2.3.1.2

Simplify.

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

Step 9.2.4

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

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Step 9.2.4.1

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

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

Step 9.2.4.2

Subtract $3$ from both sides of the equation.

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

Step 9.2.4.3

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

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Step 9.2.4.3.1

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

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

Step 9.2.4.3.2

Simplify the left side.

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Step 9.2.4.3.2.1

Dividing two negative values results in a positive value.

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

Step 9.2.4.3.2.2

Divide $x$ by $1$ .

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

Step 9.2.4.3.3

Simplify the right side.

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Step 9.2.4.3.3.1

Simplify each term.

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Step 9.2.4.3.3.1.1

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

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

Step 9.2.4.3.3.1.2

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

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

Step 9.2.4.3.3.1.3

Divide $- 3$ by $- 1$ .

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

Step 9.2.4.4

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

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

Step 9.2.4.5

Subtract $3$ from both sides of the equation.

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

Step 9.2.4.6

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

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Step 9.2.4.6.1

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

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

Step 9.2.4.6.2

Simplify the left side.

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Step 9.2.4.6.2.1

Dividing two negative values results in a positive value.

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

Step 9.2.4.6.2.2

Divide $x$ by $1$ .

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

Step 9.2.4.6.3

Simplify the right side.

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Step 9.2.4.6.3.1

Simplify each term.

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Step 9.2.4.6.3.1.1

Dividing two negative values results in a positive value.

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

Step 9.2.4.6.3.1.2

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

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

Step 9.2.4.6.3.1.3

Divide $- 3$ by $- 1$ .

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

Step 9.2.4.7

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

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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Step 10.2.1

Subtract $2$ from both sides of the equation.

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

Step 10.2.2

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

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

Step 10.2.3

Simplify the exponent.

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Step 10.2.3.1

Simplify the left side.

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Step 10.2.3.1.1

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

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Step 10.2.3.1.1.1

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

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Step 10.2.3.1.1.1.1

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

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

Step 10.2.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 10.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 10.2.3.1.1.1.2.2

Rewrite the expression.

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

Step 10.2.3.1.1.2

Simplify.

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

Step 10.2.3.2

Simplify the right side.

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Step 10.2.3.2.1

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

$3 - x = - 8$

Step 10.2.4

Solve for $x$ .

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Step 10.2.4.1

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

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Step 10.2.4.1.1

Subtract $3$ from both sides of the equation.

$- x = - 8 - 3$

Step 10.2.4.1.2

Subtract $3$ from $- 8$ .

$- x = - 11$

Step 10.2.4.2

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

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Step 10.2.4.2.1

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

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

Step 10.2.4.2.2

Simplify the left side.

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Step 10.2.4.2.2.1

Dividing two negative values results in a positive value.

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

Step 10.2.4.2.2.2

Divide $x$ by $1$ .

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

Step 10.2.4.2.3

Simplify the right side.

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Step 10.2.4.2.3.1

Divide $- 11$ by $- 1$ .

$x = 11$

Step 11

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

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Step 11.1

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

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

Step 11.2

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

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Step 11.2.1

Add $2$ to both sides of the equation.

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

Step 11.2.2

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

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

Step 11.2.3

Simplify the exponent.

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Step 11.2.3.1

Simplify the left side.

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Step 11.2.3.1.1

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

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Step 11.2.3.1.1.1

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

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Step 11.2.3.1.1.1.1

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

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

Step 11.2.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 11.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 11.2.3.1.1.1.2.2

Rewrite the expression.

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

Step 11.2.3.1.1.2

Simplify.

$3 - x = 2^{3}$

Step 11.2.3.2

Simplify the right side.

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Step 11.2.3.2.1

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

$3 - x = 8$

Step 11.2.4

Solve for $x$ .

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Step 11.2.4.1

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

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Step 11.2.4.1.1

Subtract $3$ from both sides of the equation.

$- x = 8 - 3$

Step 11.2.4.1.2

Subtract $3$ from $8$ .

$- x = 5$

Step 11.2.4.2

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

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Step 11.2.4.2.1

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

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

Step 11.2.4.2.2

Simplify the left side.

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Step 11.2.4.2.2.1

Dividing two negative values results in a positive value.

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

Step 11.2.4.2.2.2

Divide $x$ by $1$ .

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

Step 11.2.4.2.3

Simplify the right side.

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Step 11.2.4.2.3.1

Divide $5$ by $- 1$ .

$x = - 5$

Step 12

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.

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

Step 13

Exclude the solutions that do not make $4 ({(3 - x)}^{\frac{2}{3}} + 4) ({(3 - x)}^{\frac{1}{3}} + 2) ({(3 - x)}^{\frac{1}{3}} - 2) = 0$ true.

$x = 11, - 5$