Algebra Examples

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

Step 1

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

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

Step 2

Rewrite $\frac{1}{4}$ as ${(\frac{1}{2})}^{2}$ .

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

Step 3

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 $b = \frac{1}{2}$ .

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

Step 4

To write $x^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Move $2$ to the left of $x^{\frac{1}{3}}$ .

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

Step 8

To write $x^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 9

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

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Move $2$ to the left of $x^{\frac{1}{3}}$ .

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

Step 12

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

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

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

Step 13

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

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Step 13.1

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

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

Step 13.2

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

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Step 13.2.1

Set the numerator equal to zero.

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

Step 13.2.2

Solve the equation for $x$ .

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Step 13.2.2.1

Subtract $1$ from both sides of the equation.

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

Step 13.2.2.2

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

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

Step 13.2.2.3

Simplify the exponent.

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Step 13.2.2.3.1

Simplify the left side.

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Step 13.2.2.3.1.1

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

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Step 13.2.2.3.1.1.1

Apply the product rule to $2 x^{\frac{1}{3}}$ .

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

Step 13.2.2.3.1.1.2

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

$8 {(x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Step 13.2.2.3.1.1.3

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

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Step 13.2.2.3.1.1.3.1

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

$8 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 13.2.2.3.1.1.3.2

Cancel the common factor of $3$ .

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Step 13.2.2.3.1.1.3.2.1

Cancel the common factor.

$8 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 13.2.2.3.1.1.3.2.2

Rewrite the expression.

$8 x^{1} = {(- 1)}^{3}$

Step 13.2.2.3.1.1.4

Simplify.

$8 x = {(- 1)}^{3}$

Step 13.2.2.3.2

Simplify the right side.

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Step 13.2.2.3.2.1

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

$8 x = - 1$

Step 13.2.2.4

Divide each term in $8 x = - 1$ by $8$ and simplify.

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Step 13.2.2.4.1

Divide each term in $8 x = - 1$ by $8$ .

$\frac{8 x}{8} = \frac{- 1}{8}$

Step 13.2.2.4.2

Simplify the left side.

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Step 13.2.2.4.2.1

Cancel the common factor of $8$ .

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Step 13.2.2.4.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 1}{8}$

Step 13.2.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{8}$

Step 13.2.2.4.3

Simplify the right side.

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Step 13.2.2.4.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{8}$

Step 14

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

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Step 14.1

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

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

Step 14.2

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

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Step 14.2.1

Set the numerator equal to zero.

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

Step 14.2.2

Solve the equation for $x$ .

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Step 14.2.2.1

Add $1$ to both sides of the equation.

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

Step 14.2.2.2

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

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

Step 14.2.2.3

Simplify the exponent.

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Step 14.2.2.3.1

Simplify the left side.

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Step 14.2.2.3.1.1

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

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Step 14.2.2.3.1.1.1

Apply the product rule to $2 x^{\frac{1}{3}}$ .

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

Step 14.2.2.3.1.1.2

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

$8 {(x^{\frac{1}{3}})}^{3} = 1^{3}$

Step 14.2.2.3.1.1.3

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

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Step 14.2.2.3.1.1.3.1

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

$8 x^{\frac{1}{3} \cdot 3} = 1^{3}$

Step 14.2.2.3.1.1.3.2

Cancel the common factor of $3$ .

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Step 14.2.2.3.1.1.3.2.1

Cancel the common factor.

$8 x^{\frac{1}{3} \cdot 3} = 1^{3}$

Step 14.2.2.3.1.1.3.2.2

Rewrite the expression.

$8 x^{1} = 1^{3}$

Step 14.2.2.3.1.1.4

Simplify.

$8 x = 1^{3}$

Step 14.2.2.3.2

Simplify the right side.

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Step 14.2.2.3.2.1

One to any power is one.

$8 x = 1$

Step 14.2.2.4

Divide each term in $8 x = 1$ by $8$ and simplify.

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Step 14.2.2.4.1

Divide each term in $8 x = 1$ by $8$ .

$\frac{8 x}{8} = \frac{1}{8}$

Step 14.2.2.4.2

Simplify the left side.

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Step 14.2.2.4.2.1

Cancel the common factor of $8$ .

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Step 14.2.2.4.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{1}{8}$

Step 14.2.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{8}$

Step 15

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

$x = - \frac{1}{8}, \frac{1}{8}$