Solve for x (1/1024)^(x+9)=(1/256)^(-5x+5)

${(\frac{1}{1024})}^{x + 9} = {(\frac{1}{256})}^{- 5 x + 5}$

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

${(\frac{1}{4})}^{5 (x + 9)} = {(\frac{1}{4})}^{4 (- 5 x + 5)}$

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

$5 (x + 9) = 4 (- 5 x + 5)$

Solve for $x$ .

Simplify $5 (x + 9)$ .

Start simplifying.

$0 + 0 + 5 (x + 9) = 4 (- 5 x + 5)$

Simplify by adding zeros.

$5 (x + 9) = 4 (- 5 x + 5)$

Apply the distributive property.

$5 x + 5 \cdot 9 = 4 (- 5 x + 5)$

Multiply $5$ by $9$ .

$5 x + 45 = 4 (- 5 x + 5)$

Simplify $4 (- 5 x + 5)$ .

Apply the distributive property.

$5 x + 45 = 4 (- 5 x) + 4 \cdot 5$

Multiply.

Multiply $- 5$ by $4$ .

$5 x + 45 = - 20 x + 4 \cdot 5$

Multiply $4$ by $5$ .

$5 x + 45 = - 20 x + 20$

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

Add $20 x$ to both sides of the equation.

$5 x + 45 + 20 x = 20$

Add $5 x$ and $20 x$ .

$25 x + 45 = 20$

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

Subtract $45$ from both sides of the equation.

$25 x = 20 - 45$

Subtract $45$ from $20$ .

$25 x = - 25$

Divide each term in $25 x = - 25$ by $25$ and simplify.

Divide each term in $25 x = - 25$ by $25$ .

$\frac{25 x}{25} = \frac{- 25}{25}$

Simplify the left side.

Cancel the common factor of $25$ .

Cancel the common factor.

$\frac{25 x}{25} = \frac{- 25}{25}$

Divide $x$ by $1$ .

$x = \frac{- 25}{25}$

Simplify the right side.

Divide $- 25$ by $25$ .

$x = - 1$