Algebra Examples

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

Step 1

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

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

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 $\sqrt{2 + \sqrt[3]{2 x + 4}}$ as ${(2 + \sqrt[3]{2 x + 4})}^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Multiply the exponents in ${({(2 + \sqrt[3]{2 x + 4})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Factor $2$ out of $2 x + 4$ .

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Step 2.2.1.2.1

Factor $2$ out of $2 x$ .

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

Step 2.2.1.2.2

Factor $2$ out of $4$ .

${(2 + \sqrt[3]{2 x + 2 \cdot 2})}^{1} = 2^{2}$

Step 2.2.1.2.3

Factor $2$ out of $2 x + 2 \cdot 2$ .

${(2 + \sqrt[3]{2 (x + 2)})}^{1} = 2^{2}$

Step 2.2.1.3

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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

Step 3

Move all terms not containing $\sqrt[3]{2 (x + 2)}$ to the right side of the equation.

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Step 3.1

Subtract $2$ from both sides of the equation.

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

Step 3.2

Subtract $2$ from $4$ .

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

Step 4

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

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

Step 5

Simplify each side of the equation.

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

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

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Step 5.2.1.1.1

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

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

Step 5.2.1.1.2

Cancel the common factor of $3$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

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

Step 5.2.1.1.2.2

Rewrite the expression.

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

Step 5.2.1.2

Apply the distributive property.

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

Step 5.2.1.3

Multiply.

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Step 5.2.1.3.1

Multiply $2$ by $2$ .

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

Step 5.2.1.3.2

Simplify.

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

Step 5.3

Simplify the right side.

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Step 5.3.1

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

$2 x + 4 = 8$

Step 6

Solve for $x$ .

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Step 6.1

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

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Step 6.1.1

Subtract $4$ from both sides of the equation.

$2 x = 8 - 4$

Step 6.1.2

Subtract $4$ from $8$ .

$2 x = 4$

Step 6.2

Divide each term in $2 x = 4$ by $2$ and simplify.

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Step 6.2.1

Divide each term in $2 x = 4$ by $2$ .

$\frac{2 x}{2} = \frac{4}{2}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{4}{2}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{2}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $4$ by $2$ .

$x = 2$