Precalculus Examples

$81 = 27^{- x - 5}$

Step 1

Rewrite the equation as $27^{- x - 5} = 81$ .

$27^{- x - 5} = 81$

Step 2

Create equivalent expressions in the equation that all have equal bases.

$3^{3 (- x - 5)} = 3^{4}$

Step 3

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$3 (- x - 5) = 4$

Step 4

Solve for $x$ .

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Step 4.1

Divide each term in $3 (- x - 5) = 4$ by $3$ and simplify.

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Step 4.1.1

Divide each term in $3 (- x - 5) = 4$ by $3$ .

$\frac{3 (- x - 5)}{3} = \frac{4}{3}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $3$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{3 (- x - 5)}{3} = \frac{4}{3}$

Step 4.1.2.1.2

Divide $- x - 5$ by $1$ .

$- x - 5 = \frac{4}{3}$

Step 4.2

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

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Step 4.2.1

Add $5$ to both sides of the equation.

$- x = \frac{4}{3} + 5$

Step 4.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- x = \frac{4}{3} + 5 \cdot \frac{3}{3}$

Step 4.2.3

Combine $5$ and $\frac{3}{3}$ .

$- x = \frac{4}{3} + \frac{5 \cdot 3}{3}$

Step 4.2.4

Combine the numerators over the common denominator.

$- x = \frac{4 + 5 \cdot 3}{3}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $5$ by $3$ .

$- x = \frac{4 + 15}{3}$

Step 4.2.5.2

Add $4$ and $15$ .

$- x = \frac{19}{3}$

Step 4.3

Divide each term in $- x = \frac{19}{3}$ by $- 1$ and simplify.

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Step 4.3.1

Divide each term in $- x = \frac{19}{3}$ by $- 1$ .

$\frac{- x}{- 1} = \frac{\frac{19}{3}}{- 1}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{\frac{19}{3}}{- 1}$

Step 4.3.2.2

Divide $x$ by $1$ .

$x = \frac{\frac{19}{3}}{- 1}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Move the negative one from the denominator of $\frac{\frac{19}{3}}{- 1}$ .

$x = - 1 \cdot \frac{19}{3}$

Step 4.3.3.2

Rewrite $- 1 \cdot \frac{19}{3}$ as $- \frac{19}{3}$ .

$x = - \frac{19}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{19}{3}$

Decimal Form:

$x = - 6.\overline{3}$

Mixed Number Form:

$x = - 6 \frac{1}{3}$