Calculus Examples

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

Step 1

Add $1$ to both sides of the equation.

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

Step 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 3

Simplify the exponent.

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Step 3.1

Simplify the left side.

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Step 3.1.1

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

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Step 3.1.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.1.1.2

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

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

Step 3.1.1.3

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 3.1.1.4

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

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

Step 3.1.1.5

Simplify the numerator.

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Step 3.1.1.5.1

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

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Step 3.1.1.5.1.1

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

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

Step 3.1.1.5.1.2

Cancel the common factor of $3$ .

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Step 3.1.1.5.1.2.1

Cancel the common factor.

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

Step 3.1.1.5.1.2.2

Rewrite the expression.

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

Step 3.1.1.5.2

Simplify.

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

Step 3.1.1.6

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

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

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $1^{- 3}$ .

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Step 3.2.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.2.1.2

One to any power is one.

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

Step 3.2.1.3

Divide $1$ by $1$ .

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

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides of the equation by $8$ .

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

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Cancel the common factor of $8$ .

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Step 4.2.1.1.1

Cancel the common factor.

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

Step 4.2.1.1.2

Rewrite the expression.

$x = 8 \cdot 1$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Multiply $8$ by $1$ .

$x = 8$