Algebra Examples

$x - 81 x^{- 3} = 0$

Step 1

Simplify each term.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x - 81 \frac{1}{x^{3}} = 0$

Step 1.2

Combine $- 81$ and $\frac{1}{x^{3}}$ .

$x + \frac{- 81}{x^{3}} = 0$

Step 1.3

Move the negative in front of the fraction.

$x - \frac{81}{x^{3}} = 0$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x^{3}, 1$

Step 2.2

The LCM of one and any expression is the expression.

$x^{3}$

Step 3

Multiply each term in $x - \frac{81}{x^{3}} = 0$ by $x^{3}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $x - \frac{81}{x^{3}} = 0$ by $x^{3}$ .

$x \cdot x^{3} - \frac{81}{x^{3}} x^{3} = 0 x^{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 3.2.1.1.1

Multiply $x$ by $x^{3}$ .

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Step 3.2.1.1.1.1

Raise $x$ to the power of $1$ .

$x^{1} x^{3} - \frac{81}{x^{3}} x^{3} = 0 x^{3}$

Step 3.2.1.1.1.2

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

$x^{1 + 3} - \frac{81}{x^{3}} x^{3} = 0 x^{3}$

Step 3.2.1.1.2

Add $1$ and $3$ .

$x^{4} - \frac{81}{x^{3}} x^{3} = 0 x^{3}$

Step 3.2.1.2

Cancel the common factor of $x^{3}$ .

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Step 3.2.1.2.1

Move the leading negative in $- \frac{81}{x^{3}}$ into the numerator.

$x^{4} + \frac{- 81}{x^{3}} x^{3} = 0 x^{3}$

Step 3.2.1.2.2

Cancel the common factor.

$x^{4} + \frac{- 81}{x^{3}} x^{3} = 0 x^{3}$

Step 3.2.1.2.3

Rewrite the expression.

$x^{4} - 81 = 0 x^{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $0$ by $x^{3}$ .

$x^{4} - 81 = 0$

Step 4

Solve the equation.

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Step 4.1

Add $81$ to both sides of the equation.

$x^{4} = 81$

Step 4.2

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

$x = \pm \sqrt[4]{81}$

Step 4.3

Simplify $\pm \sqrt[4]{81}$ .

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Step 4.3.1

Rewrite $81$ as $3^{4}$ .

$x = \pm \sqrt[4]{3^{4}}$

Step 4.3.2

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

$x = \pm 3$

Step 4.4

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

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Step 4.4.1

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

$x = 3$

Step 4.4.2

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

$x = - 3$

Step 4.4.3

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

$x = 3, - 3$