Algebra Examples

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

Step 1

Apply the product rule to $4 x$ .

$4^{\frac{1}{3}} x^{\frac{1}{3}} - x = 0$

Step 2

Factor $x^{\frac{1}{3}}$ out of $4^{\frac{1}{3}} x^{\frac{1}{3}} - x$ .

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Step 2.1

Factor $x^{\frac{1}{3}}$ out of $4^{\frac{1}{3}} x^{\frac{1}{3}}$ .

$x^{\frac{1}{3}} \cdot 4^{\frac{1}{3}} - x = 0$

Step 2.2

Factor $x^{\frac{1}{3}}$ out of $- x$ .

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

Step 2.3

Factor $x^{\frac{1}{3}}$ out of $x^{\frac{1}{3}} \cdot 4^{\frac{1}{3}} + x^{\frac{1}{3}} (- x^{\frac{2}{3}})$ .

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

Step 3

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

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify the exponent.

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Step 4.2.2.1

Simplify the left side.

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Step 4.2.2.1.1

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

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Step 4.2.2.1.1.1

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

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Step 4.2.2.1.1.1.1

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

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

Step 4.2.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 4.2.2.1.1.1.2.1

Cancel the common factor.

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

Step 4.2.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{3}$

Step 4.2.2.1.1.2

Simplify.

$x = 0^{3}$

Step 4.2.2.2

Simplify the right side.

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Step 4.2.2.2.1

Raising $0$ to any positive power yields $0$ .

$x = 0$

Step 5

Set $4^{\frac{1}{3}} - x^{\frac{2}{3}}$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $4^{\frac{1}{3}} - x^{\frac{2}{3}}$ equal to $0$ .

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

Step 5.2

Solve $4^{\frac{1}{3}} - x^{\frac{2}{3}} = 0$ for $x$ .

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Step 5.2.1

Subtract $4^{\frac{1}{3}}$ from both sides of the equation.

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

Step 5.2.2

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

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

Step 5.2.3

Simplify the exponent.

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Step 5.2.3.1

Simplify the left side.

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Step 5.2.3.1.1

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

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Step 5.2.3.1.1.1

Apply the product rule to $- x^{\frac{2}{3}}$ .

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

Step 5.2.3.1.1.2

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

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

Step 5.2.3.1.1.3

Raise $- 1$ to the power of $3$ .

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

Step 5.2.3.1.1.4

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 5.2.3.1.1.5

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

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Step 5.2.3.1.1.5.1

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

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

Step 5.2.3.1.1.5.2

Cancel the common factor of $2$ .

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Step 5.2.3.1.1.5.2.1

Cancel the common factor.

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

Step 5.2.3.1.1.5.2.2

Rewrite the expression.

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

Step 5.2.3.1.1.5.3

Cancel the common factor of $3$ .

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Step 5.2.3.1.1.5.3.1

Cancel the common factor.

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

Step 5.2.3.1.1.5.3.2

Rewrite the expression.

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

Step 5.2.3.1.1.6

Simplify.

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

Step 5.2.3.2

Simplify the right side.

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Step 5.2.3.2.1

Simplify $\pm {(- 4^{\frac{1}{3}})}^{\frac{3}{2}}$ .

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Step 5.2.3.2.1.1

Apply the product rule to $- 4^{\frac{1}{3}}$ .

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

Step 5.2.3.2.1.2

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

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

Step 5.2.3.2.1.3

Raise $- 1$ to the power of $3$ .

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

Step 5.2.3.2.1.4

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 5.2.3.2.1.5

Simplify the expression.

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Step 5.2.3.2.1.5.1

Multiply the exponents in ${(4^{\frac{1}{3}})}^{\frac{3}{2}}$ .

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Step 5.2.3.2.1.5.1.1

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

$i x = \pm i \cdot 4^{\frac{1}{3} \cdot \frac{3}{2}}$

Step 5.2.3.2.1.5.1.2

Cancel the common factor of $3$ .

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Step 5.2.3.2.1.5.1.2.1

Cancel the common factor.

$i x = \pm i \cdot 4^{\frac{1}{3} \cdot \frac{3}{2}}$

Step 5.2.3.2.1.5.1.2.2

Rewrite the expression.

$i x = \pm i \cdot 4^{\frac{1}{2}}$

Step 5.2.3.2.1.5.2

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

$i x = \pm i \cdot {(2^{2})}^{\frac{1}{2}}$

Step 5.2.3.2.1.5.3

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

$i x = \pm i \cdot 2^{2 (\frac{1}{2})}$

Step 5.2.3.2.1.6

Cancel the common factor of $2$ .

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Step 5.2.3.2.1.6.1

Cancel the common factor.

$i x = \pm i \cdot 2^{2 (\frac{1}{2})}$

Step 5.2.3.2.1.6.2

Rewrite the expression.

$i x = \pm i \cdot 2^{1}$

Step 5.2.3.2.1.7

Evaluate the exponent.

$i x = \pm i \cdot 2$

Step 5.2.3.2.1.8

Move $2$ to the left of $i$ .

$i x = \pm 2 i$

Step 5.2.4

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

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Step 5.2.4.1

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

$i x = 2 i$

Step 5.2.4.2

Divide each term in $i x = 2 i$ by $i$ and simplify.

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Step 5.2.4.2.1

Divide each term in $i x = 2 i$ by $i$ .

$\frac{i x}{i} = \frac{2 i}{i}$

Step 5.2.4.2.2

Simplify the left side.

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Step 5.2.4.2.2.1

Cancel the common factor of $i$ .

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Step 5.2.4.2.2.1.1

Cancel the common factor.

$\frac{i x}{i} = \frac{2 i}{i}$

Step 5.2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 i}{i}$

Step 5.2.4.2.3

Simplify the right side.

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Step 5.2.4.2.3.1

Cancel the common factor of $i$ .

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Step 5.2.4.2.3.1.1

Cancel the common factor.

$x = \frac{2 i}{i}$

Step 5.2.4.2.3.1.2

Divide $2$ by $1$ .

$x = 2$

Step 5.2.4.3

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

$i x = - 2 i$

Step 5.2.4.4

Divide each term in $i x = - 2 i$ by $i$ and simplify.

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Step 5.2.4.4.1

Divide each term in $i x = - 2 i$ by $i$ .

$\frac{i x}{i} = \frac{- 2 i}{i}$

Step 5.2.4.4.2

Simplify the left side.

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Step 5.2.4.4.2.1

Cancel the common factor of $i$ .

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Step 5.2.4.4.2.1.1

Cancel the common factor.

$\frac{i x}{i} = \frac{- 2 i}{i}$

Step 5.2.4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2 i}{i}$

Step 5.2.4.4.3

Simplify the right side.

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Step 5.2.4.4.3.1

Cancel the common factor of $i$ .

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Step 5.2.4.4.3.1.1

Cancel the common factor.

$x = \frac{- 2 i}{i}$

Step 5.2.4.4.3.1.2

Divide $- 2$ by $1$ .

$x = - 2$

Step 5.2.4.5

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

$x = 2, - 2$

Step 6

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

$x = 0, 2, - 2$