Algebra Examples

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

Step 1

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

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

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

$2 x^{2} + 1 = - 8$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $1$ from both sides of the equation.

$2 x^{2} = - 8 - 1$

Step 3.1.2

Subtract $1$ from $- 8$ .

$2 x^{2} = - 9$

Step 3.2

Divide each term in $2 x^{2} = - 9$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 x^{2} = - 9$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{- 9}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{- 9}{2}$

Step 3.2.2.1.2

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

$x^{2} = \frac{- 9}{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{9}{2}$

Step 3.3

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

$x = \pm \sqrt{- \frac{9}{2}}$

Step 3.4

Simplify $\pm \sqrt{- \frac{9}{2}}$ .

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Step 3.4.1

Rewrite $- \frac{9}{2}$ as $i^{2} \frac{3^{2}}{2}$ .

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Step 3.4.1.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{9}{2}}$

Step 3.4.1.2

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{i^{2} \frac{3^{2}}{2}}$

Step 3.4.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{3^{2}}{2}}$

Step 3.4.3

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

$x = \pm i \sqrt{\frac{9}{2}}$

Step 3.4.4

Rewrite $\sqrt{\frac{9}{2}}$ as $\frac{\sqrt{9}}{\sqrt{2}}$ .

$x = \pm i \frac{\sqrt{9}}{\sqrt{2}}$

Step 3.4.5

Simplify the numerator.

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Step 3.4.5.1

Rewrite $9$ as $3^{2}$ .

$x = \pm i \frac{\sqrt{3^{2}}}{\sqrt{2}}$

Step 3.4.5.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \frac{3}{\sqrt{2}}$

Step 3.4.6

Multiply $\frac{3}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i (\frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 3.4.7

Combine and simplify the denominator.

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Step 3.4.7.1

Multiply $\frac{3}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i \frac{3 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 3.4.7.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm i \frac{3 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 3.4.7.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm i \frac{3 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 3.4.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm i \frac{3 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.4.7.5

Add $1$ and $1$ .

$x = \pm i \frac{3 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 3.4.7.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.4.7.6.1

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

$x = \pm i \frac{3 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 3.4.7.6.2

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

$x = \pm i \frac{3 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 3.4.7.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm i \frac{3 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.4.7.6.4

Cancel the common factor of $2$ .

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Step 3.4.7.6.4.1

Cancel the common factor.

$x = \pm i \frac{3 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.4.7.6.4.2

Rewrite the expression.

$x = \pm i \frac{3 \sqrt{2}}{2^{1}}$

Step 3.4.7.6.5

Evaluate the exponent.

$x = \pm i \frac{3 \sqrt{2}}{2}$

Step 3.4.8

Combine $i$ and $\frac{3 \sqrt{2}}{2}$ .

$x = \pm \frac{i (3 \sqrt{2})}{2}$

Step 3.4.9

Move $3$ to the left of $i$ .

$x = \pm \frac{3 i \sqrt{2}}{2}$

Step 3.5

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

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Step 3.5.1

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

$x = \frac{3 i \sqrt{2}}{2}$

Step 3.5.2

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

$x = - \frac{3 i \sqrt{2}}{2}$

Step 3.5.3

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

$x = \frac{3 i \sqrt{2}}{2}, - \frac{3 i \sqrt{2}}{2}$