Precalculus Examples

$\sqrt[3]{x^{3}} = \sqrt[3]{e^{4}}$

Step 1

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

${\sqrt[3]{x^{3}}}^{3} = {\sqrt[3]{e^{4}}}^{3}$

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 $\sqrt[3]{x^{3}}$ as $x^{\frac{3}{3}}$ .

${(x^{\frac{3}{3}})}^{3} = {\sqrt[3]{e^{4}}}^{3}$

Step 2.2

Divide $3$ by $3$ .

${(x^{1})}^{3} = {\sqrt[3]{e^{4}}}^{3}$

Step 2.3

Simplify the left side.

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Step 2.3.1

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

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Step 2.3.1.1

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

$x^{1 \cdot 3} = {\sqrt[3]{e^{4}}}^{3}$

Step 2.3.1.2

Multiply $3$ by $1$ .

$x^{3} = {\sqrt[3]{e^{4}}}^{3}$

Step 2.4

Simplify the right side.

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Step 2.4.1

Simplify ${\sqrt[3]{e^{4}}}^{3}$ .

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Step 2.4.1.1

Factor out $e^{3}$ .

$x^{3} = {\sqrt[3]{e^{3} e}}^{3}$

Step 2.4.1.2

Pull terms out from under the radical.

$x^{3} = {(e \sqrt[3]{e})}^{3}$

Step 2.4.1.3

Apply the product rule to $e \sqrt[3]{e}$ .

$x^{3} = e^{3} {\sqrt[3]{e}}^{3}$

Step 2.4.1.4

Rewrite ${\sqrt[3]{e}}^{3}$ as $e$ .

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Step 2.4.1.4.1

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

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

Step 2.4.1.4.2

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

$x^{3} = e^{3} e^{\frac{1}{3} \cdot 3}$

Step 2.4.1.4.3

Combine $\frac{1}{3}$ and $3$ .

$x^{3} = e^{3} e^{\frac{3}{3}}$

Step 2.4.1.4.4

Cancel the common factor of $3$ .

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Step 2.4.1.4.4.1

Cancel the common factor.

$x^{3} = e^{3} e^{\frac{3}{3}}$

Step 2.4.1.4.4.2

Rewrite the expression.

$x^{3} = e^{3} e^{1}$

Step 2.4.1.4.5

Simplify.

$x^{3} = e^{3} e$

Step 2.4.1.5

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

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Step 2.4.1.5.1

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

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Step 2.4.1.5.1.1

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

$x^{3} = e^{3} e^{1}$

Step 2.4.1.5.1.2

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

$x^{3} = e^{3 + 1}$

Step 2.4.1.5.2

Add $3$ and $1$ .

$x^{3} = e^{4}$

Step 3

Solve for $x$ .

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Step 3.1

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

$x = \sqrt[3]{e^{4}}$

Step 3.2

Simplify $\sqrt[3]{e^{4}}$ .

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Step 3.2.1

Factor out $e^{3}$ .

$x = \sqrt[3]{e^{3} e}$

Step 3.2.2

Pull terms out from under the radical.

$x = e \sqrt[3]{e}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = e \sqrt[3]{e}$

Decimal Form:

$x = 3.79366789 \dots$