Algebra Examples

${(- 3)}^{- 5} = \frac{1}{x}$

Step 1

Rewrite the equation as $\frac{1}{x} = {(- 3)}^{- 5}$ .

$\frac{1}{x} = {(- 3)}^{- 5}$

Step 2

Simplify both sides.

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Step 2.1

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

$\frac{1}{x} = \frac{1}{{(- 3)}^{5}}$

Step 2.2

Raise $- 3$ to the power of $5$ .

$\frac{1}{x} = \frac{1}{- 243}$

Step 2.3

Move the negative in front of the fraction.

$\frac{1}{x} = - \frac{1}{243}$

Step 3

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.

$1 \cdot 243 = x \cdot - 1$

Step 4

Solve the equation for $x$ .

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Step 4.1

Rewrite the equation as $x \cdot - 1 = 1 \cdot 243$ .

$x \cdot - 1 = 1 \cdot 243$

Step 4.2

Divide each term in $x \cdot - 1 = 1 \cdot 243$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $x \cdot - 1 = 1 \cdot 243$ by $- 1$ .

$\frac{x \cdot - 1}{- 1} = \frac{1 \cdot 243}{- 1}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

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

$- 1 \cdot (x \cdot - 1) = \frac{1 \cdot 243}{- 1}$

Step 4.2.2.2

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

$- (x \cdot - 1) = \frac{1 \cdot 243}{- 1}$

Step 4.2.2.3

Move $- 1$ to the left of $x$ .

$- (- 1 \cdot x) = \frac{1 \cdot 243}{- 1}$

Step 4.2.2.4

Rewrite $- 1 x$ as $- x$ .

$- - x = \frac{1 \cdot 243}{- 1}$

Step 4.2.2.5

Multiply $- - x$ .

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Step 4.2.2.5.1

Multiply $- 1$ by $- 1$ .

$1 x = \frac{1 \cdot 243}{- 1}$

Step 4.2.2.5.2

Multiply $x$ by $1$ .

$x = \frac{1 \cdot 243}{- 1}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Move the negative one from the denominator of $\frac{1 \cdot 243}{- 1}$ .

$x = - 1 \cdot (1 \cdot 243)$

Step 4.2.3.2

Rewrite $- 1 \cdot (1 \cdot 243)$ as $- (1 \cdot 243)$ .

$x = - (1 \cdot 243)$

Step 4.2.3.3

Multiply $- (1 \cdot 243)$ .

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Step 4.2.3.3.1

Multiply $243$ by $1$ .

$x = - 1 \cdot 243$

Step 4.2.3.3.2

Multiply $- 1$ by $243$ .

$x = - 243$