Solve for x (1/2)^x=2^(x+3)

${(\frac{1}{2})}^{x} = 2^{x + 3}$

Step 1

Apply the product rule to $\frac{1}{2}$ .

$\frac{1^{x}}{2^{x}} = 2^{x + 3}$

Step 2

One to any power is one.

$\frac{1}{2^{x}} = 2^{x + 3}$

Step 3

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

$2^{- x} = 2^{x + 3}$

Step 4

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

$- x = x + 3$

Step 5

Solve for $x$ .

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

Subtract $x$ from both sides of the equation.

$- x - x = 3$

Subtract $x$ from $- x$ .

$- 2 x = 3$

Divide each term in $- 2 x = 3$ by $- 2$ and simplify.

Divide each term in $- 2 x = 3$ by $- 2$ .

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 1.5$

Mixed Number Form:

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