Algebra Examples

$x^{\frac{4}{3}} = 16$

Step 1

Subtract $16$ from both sides of the equation.

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

Step 2

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

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

Step 3

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

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

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 $b = 4$ .

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Factor.

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Step 5.3.1

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

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

Step 5.3.2

Remove unnecessary parentheses.

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

Step 6

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

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

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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Step 7.2.1

Subtract $4$ from both sides of the equation.

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

Step 7.2.2

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

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

Step 7.2.3

Simplify the left side.

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Step 7.2.3.1

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

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Step 7.2.3.1.1

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

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Step 7.2.3.1.1.1

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

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

Step 7.2.3.1.1.2

Cancel the common factor of $2$ .

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Step 7.2.3.1.1.2.1

Cancel the common factor.

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

Step 7.2.3.1.1.2.2

Rewrite the expression.

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

Step 7.2.3.1.1.3

Cancel the common factor of $3$ .

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Step 7.2.3.1.1.3.1

Cancel the common factor.

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

Step 7.2.3.1.1.3.2

Rewrite the expression.

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

Step 7.2.3.1.2

Simplify.

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

Step 7.2.4

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

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Step 7.2.4.1

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

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

Step 7.2.4.2

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

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

Step 7.2.4.3

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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Step 8.2.1

Subtract $2$ from both sides of the equation.

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

Step 8.2.2

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

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

Step 8.2.3

Simplify the exponent.

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Step 8.2.3.1

Simplify the left side.

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Step 8.2.3.1.1

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

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Step 8.2.3.1.1.1

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

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Step 8.2.3.1.1.1.1

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

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

Step 8.2.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 8.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 8.2.3.1.1.1.2.2

Rewrite the expression.

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

Step 8.2.3.1.1.2

Simplify.

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

Step 8.2.3.2

Simplify the right side.

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Step 8.2.3.2.1

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

$x = - 8$

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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Step 9.2.1

Add $2$ to both sides of the equation.

$x^{\frac{1}{3}} = 2$

Step 9.2.2

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

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

Step 9.2.3

Simplify the exponent.

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Step 9.2.3.1

Simplify the left side.

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Step 9.2.3.1.1

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

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Step 9.2.3.1.1.1

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

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Step 9.2.3.1.1.1.1

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

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

Step 9.2.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 9.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 9.2.3.1.1.1.2.2

Rewrite the expression.

$x^{1} = 2^{3}$

Step 9.2.3.1.1.2

Simplify.

$x = 2^{3}$

Step 9.2.3.2

Simplify the right side.

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Step 9.2.3.2.1

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

$x = 8$

Step 10

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

$x = {(- 4)}^{\frac{3}{2}}, - {(- 4)}^{\frac{3}{2}}, - 8, 8$

Step 11

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

$x = - 8, 8$