Calculus Examples

$2 x + \frac{4}{3 \sqrt[3]{x}} = 0$

Step 1

Solve for $3 \sqrt[3]{x}$ .

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Step 1.1

Subtract $2 x$ from both sides of the equation.

$\frac{4}{3 \sqrt[3]{x}} = - 2 x$

Step 1.2

Find the LCD of the terms in the equation.

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Step 1.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$3 \sqrt[3]{x}, 1$

Step 1.2.2

The LCM of one and any expression is the expression.

$3 \sqrt[3]{x}$

Step 1.3

Multiply each term in $\frac{4}{3 \sqrt[3]{x}} = - 2 x$ by $3 \sqrt[3]{x}$ to eliminate the fractions.

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Step 1.3.1

Multiply each term in $\frac{4}{3 \sqrt[3]{x}} = - 2 x$ by $3 \sqrt[3]{x}$ .

$\frac{4}{3 \sqrt[3]{x}} (3 \sqrt[3]{x}) = - 2 x (3 \sqrt[3]{x})$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $3 \sqrt[3]{x}$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{4}{3 \sqrt[3]{x}} (3 \sqrt[3]{x}) = - 2 x (3 \sqrt[3]{x})$

Step 1.3.2.1.2

Rewrite the expression.

$4 = - 2 x (3 \sqrt[3]{x})$

Step 1.4

Solve the equation.

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Step 1.4.1

Rewrite the equation as $- 2 x (3 \sqrt[3]{x}) = 4$ .

$- 2 x (3 \sqrt[3]{x}) = 4$

Step 1.4.2

Divide each term in $- 2 x (3 \sqrt[3]{x}) = 4$ by $- 2 x$ and simplify.

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Step 1.4.2.1

Divide each term in $- 2 x (3 \sqrt[3]{x}) = 4$ by $- 2 x$ .

$\frac{- 2 x (3 \sqrt[3]{x})}{- 2 x} = \frac{4}{- 2 x}$

Step 1.4.2.2

Simplify the left side.

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Step 1.4.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.4.2.2.1.1

Cancel the common factor.

$\frac{- 2 x (3 \sqrt[3]{x})}{- 2 x} = \frac{4}{- 2 x}$

Step 1.4.2.2.1.2

Rewrite the expression.

$\frac{x (3 \sqrt[3]{x})}{x} = \frac{4}{- 2 x}$

Step 1.4.2.2.2

Cancel the common factor of $x$ .

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Step 1.4.2.2.2.1

Cancel the common factor.

$\frac{x (3 \sqrt[3]{x})}{x} = \frac{4}{- 2 x}$

Step 1.4.2.2.2.2

Divide $3 \sqrt[3]{x}$ by $1$ .

$3 \sqrt[3]{x} = \frac{4}{- 2 x}$

Step 1.4.2.3

Simplify the right side.

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Step 1.4.2.3.1

Cancel the common factor of $4$ and $- 2$ .

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Step 1.4.2.3.1.1

Factor $2$ out of $4$ .

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

Step 1.4.2.3.1.2

Cancel the common factors.

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Step 1.4.2.3.1.2.1

Factor $2$ out of $- 2 x$ .

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

Step 1.4.2.3.1.2.2

Cancel the common factor.

$3 \sqrt[3]{x} = \frac{2 \cdot 2}{2 (- x)}$

Step 1.4.2.3.1.2.3

Rewrite the expression.

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

Step 1.4.2.3.2

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

$27 {(x^{\frac{1}{3}})}^{3} = {(- \frac{2}{x})}^{3}$

Step 3.2.1.3

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

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Step 3.2.1.3.1

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

$27 x^{\frac{1}{3} \cdot 3} = {(- \frac{2}{x})}^{3}$

Step 3.2.1.3.2

Cancel the common factor of $3$ .

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Step 3.2.1.3.2.1

Cancel the common factor.

$27 x^{\frac{1}{3} \cdot 3} = {(- \frac{2}{x})}^{3}$

Step 3.2.1.3.2.2

Rewrite the expression.

$27 x^{1} = {(- \frac{2}{x})}^{3}$

Step 3.2.1.4

Simplify.

$27 x = {(- \frac{2}{x})}^{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

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

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Step 3.3.1.1.1

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

$27 x = {(- 1)}^{3} {(\frac{2}{x})}^{3}$

Step 3.3.1.1.2

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

$27 x = {(- 1)}^{3} \frac{2^{3}}{x^{3}}$

Step 3.3.1.2

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

$27 x = - \frac{2^{3}}{x^{3}}$

Step 3.3.1.3

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

$27 x = - \frac{8}{x^{3}}$

Step 4

Solve for $x$ .

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Step 4.1

Find the LCD of the terms in the equation.

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Step 4.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x^{3}$

Step 4.1.2

The LCM of one and any expression is the expression.

$x^{3}$

Step 4.2

Multiply each term in $27 x = - \frac{8}{x^{3}}$ by $x^{3}$ to eliminate the fractions.

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Step 4.2.1

Multiply each term in $27 x = - \frac{8}{x^{3}}$ by $x^{3}$ .

$27 x \cdot x^{3} = - \frac{8}{x^{3}} x^{3}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 4.2.2.1.1

Move $x^{3}$ .

$27 (x^{3} x) = - \frac{8}{x^{3}} x^{3}$

Step 4.2.2.1.2

Multiply $x^{3}$ by $x$ .

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Step 4.2.2.1.2.1

Raise $x$ to the power of $1$ .

$27 (x^{3} x^{1}) = - \frac{8}{x^{3}} x^{3}$

Step 4.2.2.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$27 x^{3 + 1} = - \frac{8}{x^{3}} x^{3}$

Step 4.2.2.1.3

Add $3$ and $1$ .

$27 x^{4} = - \frac{8}{x^{3}} x^{3}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Cancel the common factor of $x^{3}$ .

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Step 4.2.3.1.1

Move the leading negative in $- \frac{8}{x^{3}}$ into the numerator.

$27 x^{4} = \frac{- 8}{x^{3}} x^{3}$

Step 4.2.3.1.2

Cancel the common factor.

$27 x^{4} = \frac{- 8}{x^{3}} x^{3}$

Step 4.2.3.1.3

Rewrite the expression.

$27 x^{4} = - 8$

Step 4.3

Solve the equation.

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Step 4.3.1

Divide each term in $27 x^{4} = - 8$ by $27$ and simplify.

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Step 4.3.1.1

Divide each term in $27 x^{4} = - 8$ by $27$ .

$\frac{27 x^{4}}{27} = \frac{- 8}{27}$

Step 4.3.1.2

Simplify the left side.

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Step 4.3.1.2.1

Cancel the common factor of $27$ .

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Step 4.3.1.2.1.1

Cancel the common factor.

$\frac{27 x^{4}}{27} = \frac{- 8}{27}$

Step 4.3.1.2.1.2

Divide $x^{4}$ by $1$ .

$x^{4} = \frac{- 8}{27}$

Step 4.3.1.3

Simplify the right side.

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Step 4.3.1.3.1

Move the negative in front of the fraction.

$x^{4} = - \frac{8}{27}$

Step 4.3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{- \frac{8}{27}}$

Step 4.3.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.3.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt[4]{- \frac{8}{27}}$

Step 4.3.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt[4]{- \frac{8}{27}}$

Step 4.3.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt[4]{- \frac{8}{27}}, - \sqrt[4]{- \frac{8}{27}}$

Step 5

Exclude the solutions that do not make $2 x + \frac{4}{3 \sqrt[3]{x}} = 0$ true.

$No solution$