Algebra Examples

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

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 $x$ .

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Step 6.1

Simplify.

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Multiply $- 1$ by $1$ .

$2 (- x - 1) = 5$

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.1.2

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

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

Step 6.3

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

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Step 6.3.1

Add $1$ to both sides of the equation.

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

Step 6.3.2

Write $1$ as a fraction with a common denominator.

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

Step 6.3.3

Combine the numerators over the common denominator.

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

Step 6.3.4

Add $5$ and $2$ .

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

Step 6.4

Divide each term in $- x = \frac{7}{2}$ by $- 1$ and simplify.

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Step 6.4.1

Divide each term in $- x = \frac{7}{2}$ by $- 1$ .

$\frac{- x}{- 1} = \frac{\frac{7}{2}}{- 1}$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{\frac{7}{2}}{- 1}$

Step 6.4.2.2

Divide $x$ by $1$ .

$x = \frac{\frac{7}{2}}{- 1}$

Step 6.4.3

Simplify the right side.

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Step 6.4.3.1

Move the negative one from the denominator of $\frac{\frac{7}{2}}{- 1}$ .

$x = - 1 \cdot \frac{7}{2}$

Step 6.4.3.2

Rewrite $- 1 \cdot \frac{7}{2}$ as $- \frac{7}{2}$ .

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 3.5$

Mixed Number Form:

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