Evaluate (1/4)^(x+1)=32

${(\frac{1}{4})}^{x + 1} = 32$

Step 1

Apply the product rule to $\frac{1}{4}$ .

$\frac{1^{x + 1}}{4^{x + 1}} = 32$

Step 2

One to any power is one.

$\frac{1}{4^{x + 1}} = 32$

Step 3

Move $4^{x + 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$4^{- (x + 1)} = 32$

Step 4

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

$2^{2 (- (x + 1))} = 2^{5}$

Step 5

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

$2 (- (x + 1)) = 5$

Step 6

Solve for $x$ .

Simplify.

Apply the distributive property.

$2 (- x - 1 \cdot 1) = 5$

Multiply $- 1$ by $1$ .

$2 (- x - 1) = 5$

Divide each term in $2 (- x - 1) = 5$ by $- 2$ and simplify.

Divide each term in $2 (- x - 1) = 5$ by $- 2$ .

$\frac{2 (- x - 1)}{- 2} = \frac{5}{- 2}$

Simplify the left side.

Simplify the expression.

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

$- 1 \cdot (- x - 1) = \frac{5}{- 2}$

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

$- (- x - 1) = \frac{5}{- 2}$

Apply the distributive property.

$- - x - - 1 = \frac{5}{- 2}$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$1 x - - 1 = \frac{5}{- 2}$

Multiply $x$ by $1$ .

$x - - 1 = \frac{5}{- 2}$

Multiply $- 1$ by $- 1$ .

$x + 1 = \frac{5}{- 2}$

Simplify the right side.

Move the negative in front of the fraction.

$x + 1 = - \frac{5}{2}$

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

Subtract $1$ from both sides of the equation.

$x = - \frac{5}{2} - 1$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = - \frac{5}{2} - 1 \cdot \frac{2}{2}$

Combine $- 1$ and $\frac{2}{2}$ .

$x = - \frac{5}{2} + \frac{- 1 \cdot 2}{2}$

Combine the numerators over the common denominator.

$x = \frac{- 5 - 1 \cdot 2}{2}$

Simplify the numerator.

Multiply $- 1$ by $2$ .

$x = \frac{- 5 - 2}{2}$

Subtract $2$ from $- 5$ .

$x = \frac{- 7}{2}$

Move the negative in front of the fraction.

$x = - \frac{7}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{7}{2}$

Decimal Form:

$x = - 3.5$

Mixed Number Form:

$x = - 3 \frac{1}{2}$

Evaluate (1/4)^(x+1)=32

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