Precalculus Examples

${(\frac{1}{2})}^{x - 1} = 64$

Step 1

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

$\frac{1^{x - 1}}{2^{x - 1}} = 64$

Step 2

One to any power is one.

$\frac{1}{2^{x - 1}} = 64$

Step 3

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

$2^{- (x - 1)} = 64$

Step 4

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

$2^{- (x - 1)} = 2^{6}$

Step 5

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

$- (x - 1) = 6$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $- (x - 1)$ .

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Step 6.1.1

Apply the distributive property.

$- x - - 1 = 6$

Step 6.1.2

Multiply $- 1$ by $- 1$ .

$- x + 1 = 6$

Step 6.2

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

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Step 6.2.1

Subtract $1$ from both sides of the equation.

$- x = 6 - 1$

Step 6.2.2

Subtract $1$ from $6$ .

$- x = 5$

Step 6.3

Divide each term in $- x = 5$ by $- 1$ and simplify.

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Step 6.3.1

Divide each term in $- x = 5$ by $- 1$ .

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

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Dividing two negative values results in a positive value.

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

Step 6.3.2.2

Divide $x$ by $1$ .

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

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Divide $5$ by $- 1$ .

$x = - 5$