Algebra Examples

$3 \sqrt[3]{x + 4} = \sqrt[3]{2 x - 17}$

Step 1

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

${(3 \sqrt[3]{x + 4})}^{3} = {\sqrt[3]{2 x - 17}}^{3}$

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

${(3 {(x + 4)}^{\frac{1}{3}})}^{3} = {\sqrt[3]{2 x - 17}}^{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(3 {(x + 4)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1

Apply the product rule to $3 {(x + 4)}^{\frac{1}{3}}$ .

$3^{3} {({(x + 4)}^{\frac{1}{3}})}^{3} = {\sqrt[3]{2 x - 17}}^{3}$

Step 2.2.1.2

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

$27 {({(x + 4)}^{\frac{1}{3}})}^{3} = {\sqrt[3]{2 x - 17}}^{3}$

Step 2.2.1.3

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

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Step 2.2.1.3.1

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

$27 {(x + 4)}^{\frac{1}{3} \cdot 3} = {\sqrt[3]{2 x - 17}}^{3}$

Step 2.2.1.3.2

Cancel the common factor of $3$ .

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Step 2.2.1.3.2.1

Cancel the common factor.

$27 {(x + 4)}^{\frac{1}{3} \cdot 3} = {\sqrt[3]{2 x - 17}}^{3}$

Step 2.2.1.3.2.2

Rewrite the expression.

$27 {(x + 4)}^{1} = {\sqrt[3]{2 x - 17}}^{3}$

Step 2.2.1.4

Simplify.

$27 (x + 4) = {\sqrt[3]{2 x - 17}}^{3}$

Step 2.2.1.5

Apply the distributive property.

$27 x + 27 \cdot 4 = {\sqrt[3]{2 x - 17}}^{3}$

Step 2.2.1.6

Multiply $27$ by $4$ .

$27 x + 108 = {\sqrt[3]{2 x - 17}}^{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Rewrite ${\sqrt[3]{2 x - 17}}^{3}$ as $2 x - 17$ .

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Step 2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 x - 17}$ as ${(2 x - 17)}^{\frac{1}{3}}$ .

$27 x + 108 = {({(2 x - 17)}^{\frac{1}{3}})}^{3}$

Step 2.3.1.2

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

$27 x + 108 = {(2 x - 17)}^{\frac{1}{3} \cdot 3}$

Step 2.3.1.3

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

$27 x + 108 = {(2 x - 17)}^{\frac{3}{3}}$

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.

$27 x + 108 = {(2 x - 17)}^{\frac{3}{3}}$

Step 2.3.1.4.2

Rewrite the expression.

$27 x + 108 = {(2 x - 17)}^{1}$

Step 2.3.1.5

Simplify.

$27 x + 108 = 2 x - 17$

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 $2 x$ from both sides of the equation.

$27 x + 108 - 2 x = - 17$

Step 3.1.2

Subtract $2 x$ from $27 x$ .

$25 x + 108 = - 17$

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 $108$ from both sides of the equation.

$25 x = - 17 - 108$

Step 3.2.2

Subtract $108$ from $- 17$ .

$25 x = - 125$

Step 3.3

Divide each term in $25 x = - 125$ by $25$ and simplify.

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Step 3.3.1

Divide each term in $25 x = - 125$ by $25$ .

$\frac{25 x}{25} = \frac{- 125}{25}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $25$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{25 x}{25} = \frac{- 125}{25}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 125}{25}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $- 125$ by $25$ .

$x = - 5$