Solve by Factoring x^(4/3)=81

$x^{\frac{4}{3}} = 81$

Step 1

Subtract $81$ from both sides of the equation.

$x^{\frac{4}{3}} - 81 = 0$

Step 2

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

${(x^{\frac{2}{3}})}^{2} - 81 = 0$

Step 3

Rewrite $81$ as $9^{2}$ .

${(x^{\frac{2}{3}})}^{2} - 9^{2} = 0$

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{\frac{2}{3}}$ and $b = 9$ .

$(x^{\frac{2}{3}} + 9) (x^{\frac{2}{3}} - 9) = 0$

Step 5

Simplify.

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

$(x^{\frac{2}{3}} + 9) ({(x^{\frac{1}{3}})}^{2} - 9) = 0$

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

$(x^{\frac{2}{3}} + 9) ({(x^{\frac{1}{3}})}^{2} - 3^{2}) = 0$

Factor.

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

$(x^{\frac{2}{3}} + 9) ((x^{\frac{1}{3}} + 3) (x^{\frac{1}{3}} - 3)) = 0$

Remove unnecessary parentheses.

$(x^{\frac{2}{3}} + 9) (x^{\frac{1}{3}} + 3) (x^{\frac{1}{3}} - 3) = 0$

Step 6

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

$x^{\frac{2}{3}} + 9 = 0$

$x^{\frac{1}{3}} + 3 = 0$

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

Step 7

Set $x^{\frac{2}{3}} + 9$ equal to $0$ and solve for $x$ .

Set $x^{\frac{2}{3}} + 9$ equal to $0$ .

$x^{\frac{2}{3}} + 9 = 0$

Solve $x^{\frac{2}{3}} + 9 = 0$ for $x$ .

Subtract $9$ from both sides of the equation.

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

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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.

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

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

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

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

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

Step 8

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

Set $x^{\frac{1}{3}} + 3$ equal to $0$ .

$x^{\frac{1}{3}} + 3 = 0$

Solve $x^{\frac{1}{3}} + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$x^{\frac{1}{3}} = - 3$

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

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

$x = - 27$

Step 9

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

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

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

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

Add $3$ to both sides of the equation.

$x^{\frac{1}{3}} = 3$

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

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

Simplify the exponent.

Simplify the left side.

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

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

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

$x^{\frac{1}{3} \cdot 3} = 3^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 3^{3}$

Rewrite the expression.

$x^{1} = 3^{3}$

Simplify.

$x = 3^{3}$

Simplify the right side.

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

$x = 27$

Step 10

The final solution is all the values that make $(x^{\frac{2}{3}} + 9) (x^{\frac{1}{3}} + 3) (x^{\frac{1}{3}} - 3) = 0$ true.

$x = {(- 9)}^{\frac{3}{2}}, - {(- 9)}^{\frac{3}{2}}, - 27, 27$

Step 11

Exclude the solutions that do not make $(x^{\frac{2}{3}} + 9) (x^{\frac{1}{3}} + 3) (x^{\frac{1}{3}} - 3) = 0$ true.

$x = - 27, 27$

Solve by Factoring x^(4/3)=81

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