Algebra Examples

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

Step 1

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

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

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(x^{2} + 5 x)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(x^{2} + 5 x)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1.1

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

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Rewrite ${\sqrt[3]{5 x - 6}}^{3}$ as $5 x - 6$ .

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Step 2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 x - 6}$ as ${(5 x - 6)}^{\frac{1}{3}}$ .

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

Step 2.3.1.2

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

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

Step 2.3.1.3

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

$x^{2} + 5 x = {(5 x - 6)}^{\frac{3}{3}}$

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.

$x^{2} + 5 x = {(5 x - 6)}^{\frac{3}{3}}$

Step 2.3.1.4.2

Rewrite the expression.

$x^{2} + 5 x = {(5 x - 6)}^{1}$

Step 2.3.1.5

Simplify.

$x^{2} + 5 x = 5 x - 6$

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 $5 x$ from both sides of the equation.

$x^{2} + 5 x - 5 x = - 6$

Step 3.1.2

Combine the opposite terms in $x^{2} + 5 x - 5 x$ .

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Step 3.1.2.1

Subtract $5 x$ from $5 x$ .

$x^{2} + 0 = - 6$

Step 3.1.2.2

Add $x^{2}$ and $0$ .

$x^{2} = - 6$

Step 3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 6}$

Step 3.3

Simplify $\pm \sqrt{- 6}$ .

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Step 3.3.1

Rewrite $- 6$ as $- 1 (6)$ .

$x = \pm \sqrt{- 1 (6)}$

Step 3.3.2

Rewrite $\sqrt{- 1 (6)}$ as $\sqrt{- 1} \cdot \sqrt{6}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{6}$

Step 3.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \sqrt{6}$

Step 3.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i \sqrt{6}$

Step 3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i \sqrt{6}$

Step 3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i \sqrt{6}, - i \sqrt{6}$