Solve for x 32^(x+6)*1/2=8^(x-1)

$32^{x + 6} \cdot \frac{1}{2} = 8^{x - 1}$

Step 1

Raise $2$ to the power of $1$ .

$32^{x + 6} \cdot \frac{1}{2^{1}} = 8^{x - 1}$

Step 2

Move $2^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$32^{x + 6} \cdot 2^{- 1} = 8^{x - 1}$

Step 3

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

${(2^{5})}^{x + 6} \cdot 2^{- 1} = 8^{x - 1}$

Step 4

Multiply the exponents in ${(2^{5})}^{x + 6}$ .

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

$2^{5 (x + 6)} \cdot 2^{- 1} = 8^{x - 1}$

Apply the distributive property.

$2^{5 x + 5 \cdot 6} \cdot 2^{- 1} = 8^{x - 1}$

Multiply $5$ by $6$ .

$2^{5 x + 30} \cdot 2^{- 1} = 8^{x - 1}$

Step 5

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

$2^{5 x + 30 - 1} = 8^{x - 1}$

Step 6

Subtract $1$ from $30$ .

$2^{5 x + 29} = 8^{x - 1}$

Step 7

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

$2^{5 x + 29} = 2^{3 (x - 1)}$

Step 8

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

$5 x + 29 = 3 (x - 1)$

Step 9

Solve for $x$ .

Simplify $3 (x - 1)$ .

Rewrite.

$5 x + 29 = 0 + 0 + 3 (x - 1)$

Simplify by adding zeros.

$5 x + 29 = 3 (x - 1)$

Apply the distributive property.

$5 x + 29 = 3 x + 3 \cdot - 1$

Multiply $3$ by $- 1$ .

$5 x + 29 = 3 x - 3$

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

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

$5 x + 29 - 3 x = - 3$

Subtract $3 x$ from $5 x$ .

$2 x + 29 = - 3$

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

Subtract $29$ from both sides of the equation.

$2 x = - 3 - 29$

Subtract $29$ from $- 3$ .

$2 x = - 32$

Divide each term in $2 x = - 32$ by $2$ and simplify.

Divide each term in $2 x = - 32$ by $2$ .

$\frac{2 x}{2} = \frac{- 32}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 32}{2}$

Divide $x$ by $1$ .

$x = \frac{- 32}{2}$

Simplify the right side.

Divide $- 32$ by $2$ .

$x = - 16$

Solve for x 32^(x+6)*1/2=8^(x-1)

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