Algebra Examples

$\sqrt[3]{x + 5} = 2 \sqrt[3]{2 x + 6}$

Step 1

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

${\sqrt[3]{x + 5}}^{3} = {(2 \sqrt[3]{2 x + 6})}^{3}$

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 $\sqrt[3]{x + 5}$ as ${(x + 5)}^{\frac{1}{3}}$ .

${({(x + 5)}^{\frac{1}{3}})}^{3} = {(2 \sqrt[3]{2 x + 6})}^{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(x + 5)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(x + 5)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1.1

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

${(x + 5)}^{\frac{1}{3} \cdot 3} = {(2 \sqrt[3]{2 x + 6})}^{3}$

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.

${(x + 5)}^{\frac{1}{3} \cdot 3} = {(2 \sqrt[3]{2 x + 6})}^{3}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x + 5)}^{1} = {(2 \sqrt[3]{2 x + 6})}^{3}$

Step 2.2.1.2

Simplify.

$x + 5 = {(2 \sqrt[3]{2 x + 6})}^{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(2 \sqrt[3]{2 x + 6})}^{3}$ .

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Step 2.3.1.1

Factor $2$ out of $2 x + 6$ .

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Step 2.3.1.1.1

Factor $2$ out of $2 x$ .

$x + 5 = {(2 \sqrt[3]{2 (x) + 6})}^{3}$

Step 2.3.1.1.2

Factor $2$ out of $6$ .

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

Step 2.3.1.1.3

Factor $2$ out of $2 x + 2 \cdot 3$ .

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

Step 2.3.1.2

Simplify the expression.

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Step 2.3.1.2.1

Apply the product rule to $2 \sqrt[3]{2 (x + 3)}$ .

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

Step 2.3.1.2.2

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

$x + 5 = 8 {\sqrt[3]{2 (x + 3)}}^{3}$

Step 2.3.1.3

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

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Step 2.3.1.3.1

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

$x + 5 = 8 {({(2 (x + 3))}^{\frac{1}{3}})}^{3}$

Step 2.3.1.3.2

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

$x + 5 = 8 {(2 (x + 3))}^{\frac{1}{3} \cdot 3}$

Step 2.3.1.3.3

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

$x + 5 = 8 {(2 (x + 3))}^{\frac{3}{3}}$

Step 2.3.1.3.4

Cancel the common factor of $3$ .

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Step 2.3.1.3.4.1

Cancel the common factor.

$x + 5 = 8 {(2 (x + 3))}^{\frac{3}{3}}$

Step 2.3.1.3.4.2

Rewrite the expression.

$x + 5 = 8 {(2 (x + 3))}^{1}$

Step 2.3.1.3.5

Simplify.

$x + 5 = 8 (2 (x + 3))$

Step 2.3.1.4

Apply the distributive property.

$x + 5 = 8 (2 x + 2 \cdot 3)$

Step 2.3.1.5

Multiply $2$ by $3$ .

$x + 5 = 8 (2 x + 6)$

Step 2.3.1.6

Apply the distributive property.

$x + 5 = 8 (2 x) + 8 \cdot 6$

Step 2.3.1.7

Multiply.

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Step 2.3.1.7.1

Multiply $2$ by $8$ .

$x + 5 = 16 x + 8 \cdot 6$

Step 2.3.1.7.2

Multiply $8$ by $6$ .

$x + 5 = 16 x + 48$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.1.1

Subtract $16 x$ from both sides of the equation.

$x + 5 - 16 x = 48$

Step 3.1.2

Subtract $16 x$ from $x$ .

$- 15 x + 5 = 48$

Step 3.2

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

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Step 3.2.1

Subtract $5$ from both sides of the equation.

$- 15 x = 48 - 5$

Step 3.2.2

Subtract $5$ from $48$ .

$- 15 x = 43$

Step 3.3

Divide each term in $- 15 x = 43$ by $- 15$ and simplify.

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Step 3.3.1

Divide each term in $- 15 x = 43$ by $- 15$ .

$\frac{- 15 x}{- 15} = \frac{43}{- 15}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 15$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 15 x}{- 15} = \frac{43}{- 15}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{43}{- 15}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

$x = - \frac{43}{15}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{43}{15}$

Decimal Form:

$x = - 2.8\overline{6}$

Mixed Number Form:

$x = - 2 \frac{13}{15}$