Solve for x (2^(8-x))/(32^(x+12))=2^(3x-7)

$\frac{2^{8 - x}}{32^{x + 12}} = 2^{3 x - 7}$

Step 1

Move $32^{x + 12}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$2^{8 - x} \cdot 32^{- (x + 12)} = 2^{3 x - 7}$

Step 2

Rewrite $32$ as $2^{5}$ .

$2^{8 - x} \cdot {(2^{5})}^{- (x + 12)} = 2^{3 x - 7}$

Step 3

Multiply the exponents in ${(2^{5})}^{- (x + 12)}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2^{8 - x} \cdot 2^{5 (- (x + 12))} = 2^{3 x - 7}$

Apply the distributive property.

$2^{8 - x} \cdot 2^{5 (- x - 1 \cdot 12)} = 2^{3 x - 7}$

Multiply $- 1$ by $12$ .

$2^{8 - x} \cdot 2^{5 (- x - 12)} = 2^{3 x - 7}$

Apply the distributive property.

$2^{8 - x} \cdot 2^{5 (- x) + 5 \cdot - 12} = 2^{3 x - 7}$

Multiply $- 1$ by $5$ .

$2^{8 - x} \cdot 2^{- 5 x + 5 \cdot - 12} = 2^{3 x - 7}$

Multiply $5$ by $- 12$ .

$2^{8 - x} \cdot 2^{- 5 x - 60} = 2^{3 x - 7}$

Step 4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{8 - x - 5 x - 60} = 2^{3 x - 7}$

Step 5

Subtract $60$ from $8$ .

$2^{- x - 5 x - 52} = 2^{3 x - 7}$

Step 6

Subtract $5 x$ from $- x$ .

$2^{- 6 x - 52} = 2^{3 x - 7}$

Step 7

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

$- 6 x - 52 = 3 x - 7$

Step 8

Solve for $x$ .

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

Subtract $3 x$ from both sides of the equation.

$- 6 x - 52 - 3 x = - 7$

Subtract $3 x$ from $- 6 x$ .

$- 9 x - 52 = - 7$

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

Add $52$ to both sides of the equation.

$- 9 x = - 7 + 52$

Add $- 7$ and $52$ .

$- 9 x = 45$

Divide each term in $- 9 x = 45$ by $- 9$ and simplify.

Divide each term in $- 9 x = 45$ by $- 9$ .

$\frac{- 9 x}{- 9} = \frac{45}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 x}{- 9} = \frac{45}{- 9}$

Divide $x$ by $1$ .

$x = \frac{45}{- 9}$

Simplify the right side.

Divide $45$ by $- 9$ .

$x = - 5$

Solve for x (2^(8-x))/(32^(x+12))=2^(3x-7)

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