Solve for x (1/2)^(x+2)=16^(-x)

${(\frac{1}{2})}^{x + 2} = 16^{- x}$

Step 1

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

$\frac{1^{x + 2}}{2^{x + 2}} = 16^{- x}$

Step 2

One to any power is one.

$\frac{1}{2^{x + 2}} = 16^{- x}$

Step 3

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

$2^{- (x + 2)} = 16^{- x}$

Step 4

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

$2^{- (x + 2)} = 2^{4 (- x)}$

Step 5

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

$- (x + 2) = 4 (- x)$

Step 6

Solve for $x$ .

Simplify $- (x + 2)$ .

Start simplifying.

$0 + 0 - (x + 2) = 4 (- x)$

Simplify by adding zeros.

$- (x + 2) = 4 (- x)$

Apply the distributive property.

$- x - 1 \cdot 2 = 4 (- x)$

Multiply $- 1$ by $2$ .

$- x - 2 = 4 (- x)$

Multiply $- 1$ by $4$ .

$- x - 2 = - 4 x$

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

Add $4 x$ to both sides of the equation.

$- x - 2 + 4 x = 0$

Add $- x$ and $4 x$ .

$3 x - 2 = 0$

Add $2$ to both sides of the equation.

$3 x = 2$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.\overline{6}$

Solve for x (1/2)^(x+2)=16^(-x)

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