Calculus Examples

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

Step 1

Rewrite the equation as $2 x^{- \frac{1}{3}} - 2 = 0$ .

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

Step 2

Add $2$ to both sides of the equation.

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

Step 3

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

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

Step 4

Simplify the exponent.

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Step 4.1

Simplify the left side.

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Step 4.1.1

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

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Step 4.1.1.1

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

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

Step 4.1.1.2

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

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

Step 4.1.1.3

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

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

Step 4.1.1.4

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

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

Step 4.1.1.5

Simplify the numerator.

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Step 4.1.1.5.1

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

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Step 4.1.1.5.1.1

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

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

Step 4.1.1.5.1.2

Cancel the common factor of $3$ .

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Step 4.1.1.5.1.2.1

Cancel the common factor.

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

Step 4.1.1.5.1.2.2

Rewrite the expression.

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

Step 4.1.1.5.2

Simplify.

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

Step 4.1.1.6

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

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

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $2^{- 3}$ .

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 5

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = 1$