Precalculus Examples

$x^{- 6} = - \frac{1}{64}$

Step 1

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

$\frac{1}{x^{6}} = - \frac{1}{64}$

Step 2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$1 \cdot 64 = x^{6} \cdot - 1$

Step 3

Solve the equation for $x$ .

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Step 3.1

Rewrite the equation as $x^{6} \cdot - 1 = 1 \cdot 64$ .

$x^{6} \cdot - 1 = 1 \cdot 64$

Step 3.2

Divide each term in $x^{6} \cdot - 1 = 1 \cdot 64$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $x^{6} \cdot - 1 = 1 \cdot 64$ by $- 1$ .

$\frac{x^{6} \cdot - 1}{- 1} = \frac{1 \cdot 64}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Move the negative one from the denominator of $\frac{x^{6} \cdot - 1}{- 1}$ .

$- 1 \cdot (x^{6} \cdot - 1) = \frac{1 \cdot 64}{- 1}$

Step 3.2.2.2

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

$- (x^{6} \cdot - 1) = \frac{1 \cdot 64}{- 1}$

Step 3.2.2.3

Move $- 1$ to the left of $x^{6}$ .

$- (- 1 \cdot x^{6}) = \frac{1 \cdot 64}{- 1}$

Step 3.2.2.4

Rewrite $- 1 x^{6}$ as $- x^{6}$ .

$- - x^{6} = \frac{1 \cdot 64}{- 1}$

Step 3.2.2.5

Multiply $- - x^{6}$ .

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Step 3.2.2.5.1

Multiply $- 1$ by $- 1$ .

$1 x^{6} = \frac{1 \cdot 64}{- 1}$

Step 3.2.2.5.2

Multiply $x^{6}$ by $1$ .

$x^{6} = \frac{1 \cdot 64}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative one from the denominator of $\frac{1 \cdot 64}{- 1}$ .

$x^{6} = - 1 \cdot (1 \cdot 64)$

Step 3.2.3.2

Rewrite $- 1 \cdot (1 \cdot 64)$ as $- (1 \cdot 64)$ .

$x^{6} = - (1 \cdot 64)$

Step 3.2.3.3

Multiply $- (1 \cdot 64)$ .

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Step 3.2.3.3.1

Multiply $64$ by $1$ .

$x^{6} = - 1 \cdot 64$

Step 3.2.3.3.2

Multiply $- 1$ by $64$ .

$x^{6} = - 64$

Step 3.3

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

$x = \pm \sqrt[6]{- 64}$

Step 3.4

Simplify $\pm \sqrt[6]{- 64}$ .

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Step 3.4.1

Rewrite $- 64$ as ${(2 i)}^{6}$ .

$x = \pm \sqrt[6]{{(2 i)}^{6}}$

Step 3.4.2

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

$x = \pm 2 i$

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.

$x = 2 i$

Step 3.5.2

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

$x = - 2 i$

Step 3.5.3

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

$x = 2 i, - 2 i$