Precalculus Examples

${(\frac{1}{2})}^{x} = 8$

Step 1

Apply the product rule to $\frac{1}{2}$ .

$\frac{1^{x}}{2^{x}} = 8$

Step 2

One to any power is one.

$\frac{1}{2^{x}} = 8$

Step 3

Move $2^{x}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$2^{- x} = 8$

Step 4

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

$2^{- x} = 2^{3}$

Step 5

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

$- x = 3$

Step 6

Divide each term in $- x = 3$ by $- 1$ and simplify.

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Step 6.1

Divide each term in $- x = 3$ by $- 1$ .

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Dividing two negative values results in a positive value.

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

Step 6.2.2

Divide $x$ by $1$ .

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

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $3$ by $- 1$ .

$x = - 3$