Solve for x 8^(x-2)=1/16

$8^{x - 2} = \frac{1}{16}$

Step 1

Raise $16$ to the power of $1$ .

$8^{x - 2} = \frac{1}{16^{1}}$

Step 2

Move $16^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$8^{x - 2} = 16^{- 1}$

Step 3

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

$2^{3 (x - 2)} = 2^{- 4}$

Step 4

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

$3 (x - 2) = - 4$

Step 5

Solve for $x$ .

Divide each term in $3 (x - 2) = - 4$ by $3$ and simplify.

Divide each term in $3 (x - 2) = - 4$ by $3$ .

$\frac{3 (x - 2)}{3} = \frac{- 4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 (x - 2)}{3} = \frac{- 4}{3}$

Divide $x - 2$ by $1$ .

$x - 2 = \frac{- 4}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x - 2 = - \frac{4}{3}$

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

Add $2$ to both sides of the equation.

$x = - \frac{4}{3} + 2$

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = - \frac{4}{3} + 2 \cdot \frac{3}{3}$

Combine $2$ and $\frac{3}{3}$ .

$x = - \frac{4}{3} + \frac{2 \cdot 3}{3}$

Combine the numerators over the common denominator.

$x = \frac{- 4 + 2 \cdot 3}{3}$

Simplify the numerator.

Multiply $2$ by $3$ .

$x = \frac{- 4 + 6}{3}$

Add $- 4$ and $6$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.\overline{6}$

Solve for x 8^(x-2)=1/16

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