Trigonometry Examples

$64 = 32^{- x - 5}$

Step 1

Rewrite the equation as $32^{- x - 5} = 64$ .

$32^{- x - 5} = 64$

Step 2

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

$2^{5 (- x - 5)} = 2^{6}$

Step 3

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

$5 (- x - 5) = 6$

Step 4

Solve for $x$ .

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Step 4.1

Divide each term in $5 (- x - 5) = 6$ by $5$ and simplify.

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Step 4.1.1

Divide each term in $5 (- x - 5) = 6$ by $5$ .

$\frac{5 (- x - 5)}{5} = \frac{6}{5}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $5$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{5 (- x - 5)}{5} = \frac{6}{5}$

Step 4.1.2.1.2

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

$- x - 5 = \frac{6}{5}$

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.

$- x = \frac{6}{5} + 5$

Step 4.2.2

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

$- x = \frac{6}{5} + 5 \cdot \frac{5}{5}$

Step 4.2.3

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

$- x = \frac{6}{5} + \frac{5 \cdot 5}{5}$

Step 4.2.4

Combine the numerators over the common denominator.

$- x = \frac{6 + 5 \cdot 5}{5}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $5$ by $5$ .

$- x = \frac{6 + 25}{5}$

Step 4.2.5.2

Add $6$ and $25$ .

$- x = \frac{31}{5}$

Step 4.3

Divide each term in $- x = \frac{31}{5}$ by $- 1$ and simplify.

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Step 4.3.1

Divide each term in $- x = \frac{31}{5}$ by $- 1$ .

$\frac{- x}{- 1} = \frac{\frac{31}{5}}{- 1}$

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.

$\frac{x}{1} = \frac{\frac{31}{5}}{- 1}$

Step 4.3.2.2

Divide $x$ by $1$ .

$x = \frac{\frac{31}{5}}{- 1}$

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 $\frac{\frac{31}{5}}{- 1}$ .

$x = - 1 \cdot \frac{31}{5}$

Step 4.3.3.2

Rewrite $- 1 \cdot \frac{31}{5}$ as $- \frac{31}{5}$ .

$x = - \frac{31}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{31}{5}$

Decimal Form:

$x = - 6.2$

Mixed Number Form:

$x = - 6 \frac{1}{5}$