Algebra Examples

$\sqrt[3]{\frac{1}{8} - x} = - \frac{1}{2}$

Step 1

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{\frac{1}{8} - x}}^{3} = {(- \frac{1}{2})}^{3}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\frac{1}{8} - x}$ as ${(\frac{1}{8} - x)}^{\frac{1}{3}}$ .

${({(\frac{1}{8} - x)}^{\frac{1}{3}})}^{3} = {(- \frac{1}{2})}^{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Multiply the exponents in ${({(\frac{1}{8} - x)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1.1

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

${(\frac{1}{8} - x)}^{\frac{1}{3} \cdot 3} = {(- \frac{1}{2})}^{3}$

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(\frac{1}{8} - x)}^{\frac{1}{3} \cdot 3} = {(- \frac{1}{2})}^{3}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(\frac{1}{8} - x)}^{1} = {(- \frac{1}{2})}^{3}$

Step 2.2.1.2

Simplify.

$\frac{1}{8} - x = {(- \frac{1}{2})}^{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.3.1.1.1

Apply the product rule to $- \frac{1}{2}$ .

$\frac{1}{8} - x = {(- 1)}^{3} {(\frac{1}{2})}^{3}$

Step 2.3.1.1.2

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

$\frac{1}{8} - x = {(- 1)}^{3} \frac{1^{3}}{2^{3}}$

Step 2.3.1.2

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

$\frac{1}{8} - x = - \frac{1^{3}}{2^{3}}$

Step 2.3.1.3

One to any power is one.

$\frac{1}{8} - x = - \frac{1}{2^{3}}$

Step 2.3.1.4

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

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

Step 3

Solve for $x$ .

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Step 3.1

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1.1

Subtract $\frac{1}{8}$ from both sides of the equation.

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

Step 3.1.2

Combine the numerators over the common denominator.

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

Step 3.1.3

Subtract $1$ from $- 1$ .

$- x = \frac{- 2}{8}$

Step 3.1.4

Cancel the common factor of $- 2$ and $8$ .

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Step 3.1.4.1

Factor $2$ out of $- 2$ .

$- x = \frac{2 (- 1)}{8}$

Step 3.1.4.2

Cancel the common factors.

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Step 3.1.4.2.1

Factor $2$ out of $8$ .

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

Step 3.1.4.2.2

Cancel the common factor.

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

Step 3.1.4.2.3

Rewrite the expression.

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

Step 3.1.5

Move the negative in front of the fraction.

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

Step 3.2

Divide each term in $- x = - \frac{1}{4}$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- x = - \frac{1}{4}$ by $- 1$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.2.2

Divide $x$ by $1$ .

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Dividing two negative values results in a positive value.

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

Step 3.2.3.2

Divide $\frac{1}{4}$ by $1$ .

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.25$