Algebra Examples

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

Step 1

Subtract $4$ from both sides of the equation.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

Subtract $2$ from both sides of the equation.

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

Step 6.2.2

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 6.2.3

Simplify the exponent.

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Step 6.2.3.1

Simplify the left side.

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Step 6.2.3.1.1

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

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Step 6.2.3.1.1.1

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

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Step 6.2.3.1.1.1.1

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

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

Step 6.2.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 6.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 6.2.3.1.1.1.2.2

Rewrite the expression.

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

Step 6.2.3.1.1.2

Simplify.

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

Step 6.2.3.2

Simplify the right side.

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Step 6.2.3.2.1

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

$x = - 8$

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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Step 7.2.1

Add $2$ to both sides of the equation.

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

Step 7.2.2

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

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

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

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

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Step 7.2.3.1.1.1.1

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

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

Step 7.2.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 7.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 7.2.3.1.1.1.2.2

Rewrite the expression.

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

Step 7.2.3.1.1.2

Simplify.

$x = 2^{3}$

Step 7.2.3.2

Simplify the right side.

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Step 7.2.3.2.1

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

$x = 8$

Step 8

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

$x = - 8, 8$