Precalculus Examples

$\sqrt[3]{3 x - 2} = - 5$

Step 1

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

${\sqrt[3]{3 x - 2}}^{3} = {(- 5)}^{3}$

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

${({(3 x - 2)}^{\frac{1}{3}})}^{3} = {(- 5)}^{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Multiply the exponents in ${({(3 x - 2)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1.1

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

${(3 x - 2)}^{\frac{1}{3} \cdot 3} = {(- 5)}^{3}$

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.

${(3 x - 2)}^{\frac{1}{3} \cdot 3} = {(- 5)}^{3}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(3 x - 2)}^{1} = {(- 5)}^{3}$

Step 2.2.1.2

Simplify.

$3 x - 2 = {(- 5)}^{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $- 5$ to the power of $3$ .

$3 x - 2 = - 125$

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

Add $2$ to both sides of the equation.

$3 x = - 125 + 2$

Step 3.1.2

Add $- 125$ and $2$ .

$3 x = - 123$

Step 3.2

Divide each term in $3 x = - 123$ by $3$ and simplify.

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Step 3.2.1

Divide each term in $3 x = - 123$ by $3$ .

$\frac{3 x}{3} = \frac{- 123}{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 123}{3}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 123}{3}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 123$ by $3$ .

$x = - 41$