Algebra Examples

$\sqrt[3]{4 x^{2}} + 2 = x + 2$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 x^{2}}$ as ${(4 x^{2})}^{\frac{1}{3}}$ .

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

Step 2

Simplify each term.

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Step 2.1

Apply the product rule to $4 x^{2}$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Combine $2$ and $\frac{1}{3}$ .

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

Step 3

Subtract $x$ from both sides of the equation.

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

Step 4

Factor the left side of the equation.

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Step 4.1

Subtract $2$ from $2$ .

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

Step 4.2

Add $4^{\frac{1}{3}} x^{\frac{2}{3}} - x$ and $0$ .

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

Step 4.3

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

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Step 4.3.1

Factor $x^{\frac{2}{3}}$ out of $4^{\frac{1}{3}} x^{\frac{2}{3}}$ .

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

Step 4.3.2

Factor $x^{\frac{2}{3}}$ out of $- x$ .

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

Step 4.3.3

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

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

Step 5

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

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Simplify the exponent.

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Step 6.2.2.1

Simplify the left side.

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Step 6.2.2.1.1

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

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Step 6.2.2.1.1.1

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

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Step 6.2.2.1.1.1.1

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

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 0^{\frac{3}{2}}$

Step 6.2.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 6.2.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 0^{\frac{3}{2}}$

Step 6.2.2.1.1.1.2.2

Rewrite the expression.

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

Step 6.2.2.1.1.1.3

Cancel the common factor of $3$ .

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Step 6.2.2.1.1.1.3.1

Cancel the common factor.

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

Step 6.2.2.1.1.1.3.2

Rewrite the expression.

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

Step 6.2.2.1.1.2

Simplify.

$x = \pm 0^{\frac{3}{2}}$

Step 6.2.2.2

Simplify the right side.

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Step 6.2.2.2.1

Simplify $\pm 0^{\frac{3}{2}}$ .

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Step 6.2.2.2.1.1

Simplify the expression.

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Step 6.2.2.2.1.1.1

Rewrite $0$ as $0^{2}$ .

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

Step 6.2.2.2.1.1.2

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

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

Step 6.2.2.2.1.2

Cancel the common factor of $2$ .

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Step 6.2.2.2.1.2.1

Cancel the common factor.

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

Step 6.2.2.2.1.2.2

Rewrite the expression.

$x = \pm 0^{3}$

Step 6.2.2.2.1.3

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

$x = \pm 0$

Step 6.2.2.2.1.4

Plus or minus $0$ is $0$ .

$x = 0$

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.2.3

Simplify the exponent.

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Step 7.2.3.1

Simplify the left side.

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Step 7.2.3.1.1

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

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Step 7.2.3.1.1.1

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

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

Step 7.2.3.1.1.2

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

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

Step 7.2.3.1.1.3

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

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Step 7.2.3.1.1.3.1

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

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

Step 7.2.3.1.1.3.2

Cancel the common factor of $3$ .

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Step 7.2.3.1.1.3.2.1

Cancel the common factor.

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

Step 7.2.3.1.1.3.2.2

Rewrite the expression.

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

Step 7.2.3.1.1.4

Simplify.

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

Step 7.2.3.2

Simplify the right side.

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Step 7.2.3.2.1

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

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Step 7.2.3.2.1.1

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

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

Step 7.2.3.2.1.2

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

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

Step 7.2.3.2.1.3

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

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Step 7.2.3.2.1.3.1

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

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

Step 7.2.3.2.1.3.2

Cancel the common factor of $3$ .

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Step 7.2.3.2.1.3.2.1

Cancel the common factor.

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

Step 7.2.3.2.1.3.2.2

Rewrite the expression.

$- x = - 4^{1}$

Step 7.2.3.2.1.4

Evaluate the exponent.

$- x = - 1 \cdot 4$

Step 7.2.3.2.1.5

Multiply $- 1$ by $4$ .

$- x = - 4$

Step 7.2.4

Divide each term in $- x = - 4$ by $- 1$ and simplify.

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Step 7.2.4.1

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Step 7.2.4.2

Simplify the left side.

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Step 7.2.4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Step 7.2.4.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Step 7.2.4.3

Simplify the right side.

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Step 7.2.4.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 8

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

$x = 0, 4$