Algebra Examples

$\sqrt[3]{\frac{1}{4} x + 3} = 2$

Step 1

Subtract $2$ from both sides of the equation.

$\sqrt[3]{\frac{1}{4} x + 3} - 2 = 0$

Step 2

Simplify $\sqrt[3]{\frac{1}{4} x + 3} - 2$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $\frac{1}{4}$ and $x$ .

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

Step 2.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 2.1.3

Combine $3$ and $\frac{4}{4}$ .

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

Step 2.1.4

Combine the numerators over the common denominator.

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

Step 2.1.5

Multiply $3$ by $4$ .

$\sqrt[3]{\frac{x + 12}{4}} - 2 = 0$

Step 2.1.6

Rewrite $\sqrt[3]{\frac{x + 12}{4}}$ as $\frac{\sqrt[3]{x + 12}}{\sqrt[3]{4}}$ .

$\frac{\sqrt[3]{x + 12}}{\sqrt[3]{4}} - 2 = 0$

Step 2.1.7

Multiply $\frac{\sqrt[3]{x + 12}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$\frac{\sqrt[3]{x + 12}}{\sqrt[3]{4}} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}} - 2 = 0$

Step 2.1.8

Combine and simplify the denominator.

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Step 2.1.8.1

Multiply $\frac{\sqrt[3]{x + 12}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$\frac{\sqrt[3]{x + 12} {\sqrt[3]{4}}^{2}}{\sqrt[3]{4} {\sqrt[3]{4}}^{2}} - 2 = 0$

Step 2.1.8.2

Raise $\sqrt[3]{4}$ to the power of $1$ .

$\frac{\sqrt[3]{x + 12} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2}} - 2 = 0$

Step 2.1.8.3

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

$\frac{\sqrt[3]{x + 12} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}} - 2 = 0$

Step 2.1.8.4

Add $1$ and $2$ .

$\frac{\sqrt[3]{x + 12} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{3}} - 2 = 0$

Step 2.1.8.5

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

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Step 2.1.8.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{x + 12} {\sqrt[3]{4}}^{2}}{{(4^{\frac{1}{3}})}^{3}} - 2 = 0$

Step 2.1.8.5.2

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

$\frac{\sqrt[3]{x + 12} {\sqrt[3]{4}}^{2}}{4^{\frac{1}{3} \cdot 3}} - 2 = 0$

Step 2.1.8.5.3

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

$\frac{\sqrt[3]{x + 12} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}} - 2 = 0$

Step 2.1.8.5.4

Cancel the common factor of $3$ .

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Step 2.1.8.5.4.1

Cancel the common factor.

$\frac{\sqrt[3]{x + 12} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}} - 2 = 0$

Step 2.1.8.5.4.2

Rewrite the expression.

$\frac{\sqrt[3]{x + 12} {\sqrt[3]{4}}^{2}}{4^{1}} - 2 = 0$

Step 2.1.8.5.5

Evaluate the exponent.

$\frac{\sqrt[3]{x + 12} {\sqrt[3]{4}}^{2}}{4} - 2 = 0$

Step 2.1.9

Simplify the numerator.

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Step 2.1.9.1

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

$\frac{\sqrt[3]{x + 12} \sqrt[3]{4^{2}}}{4} - 2 = 0$

Step 2.1.9.2

Raise $4$ to the power of $2$ .

$\frac{\sqrt[3]{x + 12} \sqrt[3]{16}}{4} - 2 = 0$

Step 2.1.9.3

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 2.1.9.3.1

Factor $8$ out of $16$ .

$\frac{\sqrt[3]{x + 12} \sqrt[3]{8 (2)}}{4} - 2 = 0$

Step 2.1.9.3.2

Rewrite $8$ as $2^{3}$ .

$\frac{\sqrt[3]{x + 12} \sqrt[3]{2^{3} \cdot 2}}{4} - 2 = 0$

Step 2.1.9.4

Pull terms out from under the radical.

$\frac{\sqrt[3]{x + 12} \cdot 2 \sqrt[3]{2}}{4} - 2 = 0$

Step 2.1.9.5

Combine using the product rule for radicals.

$\frac{2 \sqrt[3]{(x + 12) \cdot 2}}{4} - 2 = 0$

Step 2.1.10

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

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Step 2.1.10.1

Factor $2$ out of $2 \sqrt[3]{(x + 12) \cdot 2}$ .

$\frac{2 (\sqrt[3]{(x + 12) \cdot 2})}{4} - 2 = 0$

Step 2.1.10.2

Cancel the common factors.

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Step 2.1.10.2.1

Factor $2$ out of $4$ .

$\frac{2 \sqrt[3]{(x + 12) \cdot 2}}{2 \cdot 2} - 2 = 0$

Step 2.1.10.2.2

Cancel the common factor.

$\frac{2 \sqrt[3]{(x + 12) \cdot 2}}{2 \cdot 2} - 2 = 0$

Step 2.1.10.2.3

Rewrite the expression.

$\frac{\sqrt[3]{(x + 12) \cdot 2}}{2} - 2 = 0$

Step 2.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\sqrt[3]{(x + 12) \cdot 2}}{2} - 2 \cdot \frac{2}{2} = 0$

Step 2.3

Combine $- 2$ and $\frac{2}{2}$ .

$\frac{\sqrt[3]{(x + 12) \cdot 2}}{2} + \frac{- 2 \cdot 2}{2} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{\sqrt[3]{(x + 12) \cdot 2} - 2 \cdot 2}{2} = 0$

Step 2.5

Multiply $- 2$ by $2$ .

$\frac{\sqrt[3]{(x + 12) \cdot 2} - 4}{2} = 0$

Step 2.6

Reorder factors in $\frac{\sqrt[3]{(x + 12) \cdot 2} - 4}{2}$ .

$\frac{\sqrt[3]{2 (x + 12)} - 4}{2} = 0$

Step 3

Set the numerator equal to zero.

$\sqrt[3]{2 (x + 12)} - 4 = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Add $4$ to both sides of the equation.

$\sqrt[3]{2 (x + 12)} = 4$

Step 4.2

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

${\sqrt[3]{2 (x + 12)}}^{3} = 4^{3}$

Step 4.3

Simplify each side of the equation.

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Step 4.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 (x + 12)}$ as ${(2 (x + 12))}^{\frac{1}{3}}$ .

${({(2 (x + 12))}^{\frac{1}{3}})}^{3} = 4^{3}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Simplify ${({(2 (x + 12))}^{\frac{1}{3}})}^{3}$ .

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Step 4.3.2.1.1

Multiply the exponents in ${({(2 (x + 12))}^{\frac{1}{3}})}^{3}$ .

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Step 4.3.2.1.1.1

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

${(2 (x + 12))}^{\frac{1}{3} \cdot 3} = 4^{3}$

Step 4.3.2.1.1.2

Cancel the common factor of $3$ .

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Step 4.3.2.1.1.2.1

Cancel the common factor.

${(2 (x + 12))}^{\frac{1}{3} \cdot 3} = 4^{3}$

Step 4.3.2.1.1.2.2

Rewrite the expression.

${(2 (x + 12))}^{1} = 4^{3}$

Step 4.3.2.1.2

Apply the distributive property.

${(2 x + 2 \cdot 12)}^{1} = 4^{3}$

Step 4.3.2.1.3

Multiply.

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Step 4.3.2.1.3.1

Multiply $2$ by $12$ .

${(2 x + 24)}^{1} = 4^{3}$

Step 4.3.2.1.3.2

Simplify.

$2 x + 24 = 4^{3}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Raise $4$ to the power of $3$ .

$2 x + 24 = 64$

Step 4.4

Solve for $x$ .

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Step 4.4.1

Move all terms not containing $x$ to the right side of the equation.

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Step 4.4.1.1

Subtract $24$ from both sides of the equation.

$2 x = 64 - 24$

Step 4.4.1.2

Subtract $24$ from $64$ .

$2 x = 40$

Step 4.4.2

Divide each term in $2 x = 40$ by $2$ and simplify.

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Step 4.4.2.1

Divide each term in $2 x = 40$ by $2$ .

$\frac{2 x}{2} = \frac{40}{2}$

Step 4.4.2.2

Simplify the left side.

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Step 4.4.2.2.1

Cancel the common factor of $2$ .

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Step 4.4.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{40}{2}$

Step 4.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{40}{2}$

Step 4.4.2.3

Simplify the right side.

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Step 4.4.2.3.1

Divide $40$ by $2$ .

$x = 20$