Algebra Examples

$4^{x} = {(\frac{1}{8})}^{x + 5}$

Step 1

Apply the product rule to $\frac{1}{8}$ .

$4^{x} = \frac{1^{x + 5}}{8^{x + 5}}$

Step 2

One to any power is one.

$4^{x} = \frac{1}{8^{x + 5}}$

Step 3

Move $8^{x + 5}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$4^{x} = 8^{- (x + 5)}$

Step 4

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

$2^{2 x} = 2^{3 (- (x + 5))}$

Step 5

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

$2 x = 3 (- (x + 5))$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $3 (- (x + 5))$ .

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Step 6.1.1

Apply the distributive property.

$2 x = 3 (- x - 1 \cdot 5)$

Step 6.1.2

Multiply $- 1$ by $5$ .

$2 x = 3 (- x - 5)$

Step 6.1.3

Apply the distributive property.

$2 x = 3 (- x) + 3 \cdot - 5$

Step 6.1.4

Multiply.

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Step 6.1.4.1

Multiply $- 1$ by $3$ .

$2 x = - 3 x + 3 \cdot - 5$

Step 6.1.4.2

Multiply $3$ by $- 5$ .

$2 x = - 3 x - 15$

Step 6.2

Move all terms containing $x$ to the left side of the equation.

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Step 6.2.1

Add $3 x$ to both sides of the equation.

$2 x + 3 x = - 15$

Step 6.2.2

Add $2 x$ and $3 x$ .

$5 x = - 15$

Step 6.3

Divide each term in $5 x = - 15$ by $5$ and simplify.

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Step 6.3.1

Divide each term in $5 x = - 15$ by $5$ .

$\frac{5 x}{5} = \frac{- 15}{5}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $5$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 15}{5}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 15}{5}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Divide $- 15$ by $5$ .

$x = - 3$