Algebra Examples

$5 - \sqrt[3]{x^{2} - 9} = 7$

Step 1

Move all terms not containing $- \sqrt[3]{x^{2} - 9}$ to the right side of the equation.

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Step 1.1

Subtract $5$ from both sides of the equation.

$- \sqrt[3]{x^{2} - 9} = 7 - 5$

Step 1.2

Subtract $5$ from $7$ .

$- \sqrt[3]{x^{2} - 9} = 2$

Step 2

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

${(- \sqrt[3]{x^{2} - 9})}^{3} = 2^{3}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2} - 9}$ as ${(x^{2} - 9)}^{\frac{1}{3}}$ .

${(- {(x^{2} - 9)}^{\frac{1}{3}})}^{3} = 2^{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(- {(x^{2} - 9)}^{\frac{1}{3}})}^{3}$ .

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Step 3.2.1.1

Apply the product rule to $- {(x^{2} - 9)}^{\frac{1}{3}}$ .

${(- 1)}^{3} {({(x^{2} - 9)}^{\frac{1}{3}})}^{3} = 2^{3}$

Step 3.2.1.2

Raise $- 1$ to the power of $3$ .

$- {({(x^{2} - 9)}^{\frac{1}{3}})}^{3} = 2^{3}$

Step 3.2.1.3

Multiply the exponents in ${({(x^{2} - 9)}^{\frac{1}{3}})}^{3}$ .

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Step 3.2.1.3.1

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

$- {(x^{2} - 9)}^{\frac{1}{3} \cdot 3} = 2^{3}$

Step 3.2.1.3.2

Cancel the common factor of $3$ .

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Step 3.2.1.3.2.1

Cancel the common factor.

$- {(x^{2} - 9)}^{\frac{1}{3} \cdot 3} = 2^{3}$

Step 3.2.1.3.2.2

Rewrite the expression.

$- {(x^{2} - 9)}^{1} = 2^{3}$

Step 3.2.1.4

Simplify.

$- (x^{2} - 9) = 2^{3}$

Step 3.2.1.5

Apply the distributive property.

$- x^{2} - - 9 = 2^{3}$

Step 3.2.1.6

Multiply $- 1$ by $- 9$ .

$- x^{2} + 9 = 2^{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

$- x^{2} + 9 = 8$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Subtract $9$ from both sides of the equation.

$- x^{2} = 8 - 9$

Step 4.1.2

Subtract $9$ from $8$ .

$- x^{2} = - 1$

Step 4.2

Divide each term in $- x^{2} = - 1$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $- x^{2} = - 1$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 1}{- 1}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 1}{- 1}$

Step 4.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 1}{- 1}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $- 1$ by $- 1$ .

$x^{2} = 1$

Step 4.3

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

$x = \pm \sqrt{1}$

Step 4.4

Any root of $1$ is $1$ .

$x = \pm 1$

Step 4.5

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

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Step 4.5.1

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

$x = 1$

Step 4.5.2

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

$x = - 1$

Step 4.5.3

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

$x = 1, - 1$