Finite Math Examples

$\frac{\sqrt[3]{x} - \sqrt[9]{x}}{3 \sqrt[3]{x} + \sqrt[9]{x}}$

Step 1

Set the denominator in $\frac{\sqrt[3]{x} - \sqrt[9]{x}}{3 \sqrt[3]{x} + \sqrt[9]{x}}$ equal to $0$ to find where the expression is undefined.

$3 \sqrt[3]{x} + \sqrt[9]{x} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $\sqrt[9]{x}$ from both sides of the equation.

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

Step 2.2

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

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

Step 2.3

Simplify each side of the equation.

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Step 2.3.1

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

${(3 x^{\frac{1}{3}})}^{3} = {(- \sqrt[9]{x})}^{3}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

Apply the product rule to $3 x^{\frac{1}{3}}$ .

$3^{3} {(x^{\frac{1}{3}})}^{3} = {(- \sqrt[9]{x})}^{3}$

Step 2.3.2.1.2

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

$27 {(x^{\frac{1}{3}})}^{3} = {(- \sqrt[9]{x})}^{3}$

Step 2.3.2.1.3

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

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Step 2.3.2.1.3.1

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

$27 x^{\frac{1}{3} \cdot 3} = {(- \sqrt[9]{x})}^{3}$

Step 2.3.2.1.3.2

Cancel the common factor of $3$ .

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Step 2.3.2.1.3.2.1

Cancel the common factor.

$27 x^{\frac{1}{3} \cdot 3} = {(- \sqrt[9]{x})}^{3}$

Step 2.3.2.1.3.2.2

Rewrite the expression.

$27 x^{1} = {(- \sqrt[9]{x})}^{3}$

Step 2.3.2.1.4

Simplify.

$27 x = {(- \sqrt[9]{x})}^{3}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify ${(- \sqrt[9]{x})}^{3}$ .

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Step 2.3.3.1.1

Simplify the expression.

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Step 2.3.3.1.1.1

Apply the product rule to $- \sqrt[9]{x}$ .

$27 x = {(- 1)}^{3} {\sqrt[9]{x}}^{3}$

Step 2.3.3.1.1.2

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

$27 x = - {\sqrt[9]{x}}^{3}$

Step 2.3.3.1.2

Rewrite ${\sqrt[9]{x}}^{3}$ as $\sqrt[3]{x}$ .

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Step 2.3.3.1.2.1

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

$27 x = - {(x^{\frac{1}{9}})}^{3}$

Step 2.3.3.1.2.2

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

$27 x = - x^{\frac{1}{9} \cdot 3}$

Step 2.3.3.1.2.3

Combine $\frac{1}{9}$ and $3$ .

$27 x = - x^{\frac{3}{9}}$

Step 2.3.3.1.2.4

Cancel the common factor of $3$ and $9$ .

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Step 2.3.3.1.2.4.1

Factor $3$ out of $3$ .

$27 x = - x^{\frac{3 (1)}{9}}$

Step 2.3.3.1.2.4.2

Cancel the common factors.

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Step 2.3.3.1.2.4.2.1

Factor $3$ out of $9$ .

$27 x = - x^{\frac{3 \cdot 1}{3 \cdot 3}}$

Step 2.3.3.1.2.4.2.2

Cancel the common factor.

$27 x = - x^{\frac{3 \cdot 1}{3 \cdot 3}}$

Step 2.3.3.1.2.4.2.3

Rewrite the expression.

$27 x = - x^{\frac{1}{3}}$

Step 2.3.3.1.2.5

Rewrite $x^{\frac{1}{3}}$ as $\sqrt[3]{x}$ .

$27 x = - \sqrt[3]{x}$

Step 2.4

Rewrite the equation as $- \sqrt[3]{x} = 27 x$ .

$- \sqrt[3]{x} = 27 x$

Step 2.5

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

${(- \sqrt[3]{x})}^{3} = {(27 x)}^{3}$

Step 2.6

Simplify each side of the equation.

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Step 2.6.1

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

${(- x^{\frac{1}{3}})}^{3} = {(27 x)}^{3}$

Step 2.6.2

Simplify the left side.

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Step 2.6.2.1

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

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Step 2.6.2.1.1

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

${(- 1)}^{3} {(x^{\frac{1}{3}})}^{3} = {(27 x)}^{3}$

Step 2.6.2.1.2

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

$- {(x^{\frac{1}{3}})}^{3} = {(27 x)}^{3}$

Step 2.6.2.1.3

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

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Step 2.6.2.1.3.1

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

$- x^{\frac{1}{3} \cdot 3} = {(27 x)}^{3}$

Step 2.6.2.1.3.2

Cancel the common factor of $3$ .

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Step 2.6.2.1.3.2.1

Cancel the common factor.

$- x^{\frac{1}{3} \cdot 3} = {(27 x)}^{3}$

Step 2.6.2.1.3.2.2

Rewrite the expression.

$- x^{1} = {(27 x)}^{3}$

Step 2.6.2.1.4

Simplify.

$- x = {(27 x)}^{3}$

Step 2.6.3

Simplify the right side.

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Step 2.6.3.1

Simplify ${(27 x)}^{3}$ .

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Step 2.6.3.1.1

Apply the product rule to $27 x$ .

$- x = 27^{3} x^{3}$

Step 2.6.3.1.2

Raise $27$ to the power of $3$ .

$- x = 19683 x^{3}$

Step 2.7

Solve for $x$ .

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Step 2.7.1

Subtract $19683 x^{3}$ from both sides of the equation.

$- x - 19683 x^{3} = 0$

Step 2.7.2

Factor $- x$ out of $- x - 19683 x^{3}$ .

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Step 2.7.2.1

Reorder $- x$ and $- 19683 x^{3}$ .

$- 19683 x^{3} - x = 0$

Step 2.7.2.2

Factor $- x$ out of $- 19683 x^{3}$ .

$- x (19683 x^{2}) - x = 0$

Step 2.7.2.3

Factor $- x$ out of $- x$ .

$- x (19683 x^{2}) - x \cdot 1 = 0$

Step 2.7.2.4

Factor $- x$ out of $- x (19683 x^{2}) - x (1)$ .

$- x (19683 x^{2} + 1) = 0$

Step 2.7.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$19683 x^{2} + 1 = 0$

Step 2.7.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.7.5

Set $19683 x^{2} + 1$ equal to $0$ and solve for $x$ .

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Step 2.7.5.1

Set $19683 x^{2} + 1$ equal to $0$ .

$19683 x^{2} + 1 = 0$

Step 2.7.5.2

Solve $19683 x^{2} + 1 = 0$ for $x$ .

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Step 2.7.5.2.1

Subtract $1$ from both sides of the equation.

$19683 x^{2} = - 1$

Step 2.7.5.2.2

Divide each term in $19683 x^{2} = - 1$ by $19683$ and simplify.

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Step 2.7.5.2.2.1

Divide each term in $19683 x^{2} = - 1$ by $19683$ .

$\frac{19683 x^{2}}{19683} = \frac{- 1}{19683}$

Step 2.7.5.2.2.2

Simplify the left side.

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Step 2.7.5.2.2.2.1

Cancel the common factor of $19683$ .

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Step 2.7.5.2.2.2.1.1

Cancel the common factor.

$\frac{19683 x^{2}}{19683} = \frac{- 1}{19683}$

Step 2.7.5.2.2.2.1.2

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

$x^{2} = \frac{- 1}{19683}$

Step 2.7.5.2.2.3

Simplify the right side.

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Step 2.7.5.2.2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{1}{19683}$

Step 2.7.5.2.3

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

$x = \pm \sqrt{- \frac{1}{19683}}$

Step 2.7.5.2.4

Simplify $\pm \sqrt{- \frac{1}{19683}}$ .

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Step 2.7.5.2.4.1

Rewrite $- \frac{1}{19683}$ as ${(\frac{i}{81})}^{2} \frac{1}{3}$ .

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Step 2.7.5.2.4.1.1

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

$x = \pm \sqrt{i^{2} \frac{1}{19683}}$

Step 2.7.5.2.4.1.2

Factor the perfect power $1^{2}$ out of $1$ .

$x = \pm \sqrt{i^{2} \frac{1^{2} \cdot 1}{19683}}$

Step 2.7.5.2.4.1.3

Factor the perfect power $81^{2}$ out of $19683$ .

$x = \pm \sqrt{i^{2} \frac{1^{2} \cdot 1}{81^{2} \cdot 3}}$

Step 2.7.5.2.4.1.4

Rearrange the fraction $\frac{1^{2} \cdot 1}{81^{2} \cdot 3}$ .

$x = \pm \sqrt{i^{2} ({(\frac{1}{81})}^{2} \frac{1}{3})}$

Step 2.7.5.2.4.1.5

Rewrite $i^{2} {(\frac{1}{81})}^{2}$ as ${(\frac{i}{81})}^{2}$ .

$x = \pm \sqrt{{(\frac{i}{81})}^{2} \frac{1}{3}}$

Step 2.7.5.2.4.2

Pull terms out from under the radical.

$x = \pm \frac{i}{81} \sqrt{\frac{1}{3}}$

Step 2.7.5.2.4.3

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$x = \pm \frac{i}{81} \cdot \frac{\sqrt{1}}{\sqrt{3}}$

Step 2.7.5.2.4.4

Any root of $1$ is $1$ .

$x = \pm \frac{i}{81} \cdot \frac{1}{\sqrt{3}}$

Step 2.7.5.2.4.5

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{i}{81} (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 2.7.5.2.4.6

Combine and simplify the denominator.

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Step 2.7.5.2.4.6.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{i}{81} \cdot \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.7.5.2.4.6.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{i}{81} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.7.5.2.4.6.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{i}{81} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.7.5.2.4.6.4

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

$x = \pm \frac{i}{81} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.7.5.2.4.6.5

Add $1$ and $1$ .

$x = \pm \frac{i}{81} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.7.5.2.4.6.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.7.5.2.4.6.6.1

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

$x = \pm \frac{i}{81} \cdot \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.7.5.2.4.6.6.2

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

$x = \pm \frac{i}{81} \cdot \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.7.5.2.4.6.6.3

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

$x = \pm \frac{i}{81} \cdot \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.7.5.2.4.6.6.4

Cancel the common factor of $2$ .

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Step 2.7.5.2.4.6.6.4.1

Cancel the common factor.

$x = \pm \frac{i}{81} \cdot \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.7.5.2.4.6.6.4.2

Rewrite the expression.

$x = \pm \frac{i}{81} \cdot \frac{\sqrt{3}}{3^{1}}$

Step 2.7.5.2.4.6.6.5

Evaluate the exponent.

$x = \pm \frac{i}{81} \cdot \frac{\sqrt{3}}{3}$

Step 2.7.5.2.4.7

Multiply $\frac{i}{81} \cdot \frac{\sqrt{3}}{3}$ .

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Step 2.7.5.2.4.7.1

Multiply $\frac{i}{81}$ by $\frac{\sqrt{3}}{3}$ .

$x = \pm \frac{i \sqrt{3}}{81 \cdot 3}$

Step 2.7.5.2.4.7.2

Multiply $81$ by $3$ .

$x = \pm \frac{i \sqrt{3}}{243}$

Step 2.7.5.2.5

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

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Step 2.7.5.2.5.1

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

$x = \frac{i \sqrt{3}}{243}$

Step 2.7.5.2.5.2

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

$x = - \frac{i \sqrt{3}}{243}$

Step 2.7.5.2.5.3

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

$x = \frac{i \sqrt{3}}{243}, - \frac{i \sqrt{3}}{243}$

Step 2.7.6

The final solution is all the values that make $- x (19683 x^{2} + 1) = 0$ true.

$x = 0, \frac{i \sqrt{3}}{243}, - \frac{i \sqrt{3}}{243}$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 4