Algebra Examples

$\sqrt[3]{2 x + 1} = \sqrt[3]{x + 8}$

Step 1

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

${\sqrt[3]{2 x + 1}}^{3} = {\sqrt[3]{x + 8}}^{3}$

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

${({(2 x + 1)}^{\frac{1}{3}})}^{3} = {\sqrt[3]{x + 8}}^{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

${(2 x + 1)}^{1} = {\sqrt[3]{x + 8}}^{3}$

Step 2.2.1.2

Simplify.

$2 x + 1 = {\sqrt[3]{x + 8}}^{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Rewrite ${\sqrt[3]{x + 8}}^{3}$ as $x + 8$ .

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Step 2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x + 8}$ as ${(x + 8)}^{\frac{1}{3}}$ .

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

Step 2.3.1.2

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

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

Step 2.3.1.3

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

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

Step 2.3.1.4

Cancel the common factor of $3$ .

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Step 2.3.1.4.1

Cancel the common factor.

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

Step 2.3.1.4.2

Rewrite the expression.

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

Step 2.3.1.5

Simplify.

$2 x + 1 = x + 8$

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 $x$ from both sides of the equation.

$2 x + 1 - x = 8$

Step 3.1.2

Subtract $x$ from $2 x$ .

$x + 1 = 8$

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 $1$ from both sides of the equation.

$x = 8 - 1$

Step 3.2.2

Subtract $1$ from $8$ .

$x = 7$