Solve for x 216=(1/36)^(x-1)

$216 = {(\frac{1}{36})}^{x - 1}$

Rewrite the equation as ${(\frac{1}{36})}^{x - 1} = 216$ .

${(\frac{1}{36})}^{x - 1} = 216$

Apply the product rule to $\frac{1}{36}$ .

$\frac{1^{x - 1}}{36^{x - 1}} = 216$

One to any power is one.

$\frac{1}{36^{x - 1}} = 216$

Move $36^{x - 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$36^{- (x - 1)} = 216$

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

$6^{2 (- (x - 1))} = 6^{3}$

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

$2 (- (x - 1)) = 3$

Solve for $x$ .

Simplify.

Apply the distributive property.

$2 (- x - - 1) = 3$

Multiply $- 1$ by $- 1$ .

$2 (- x + 1) = 3$

Divide each term in $2 (- x + 1) = 3$ by $- 2$ and simplify.

Divide each term in $2 (- x + 1) = 3$ by $- 2$ .

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

Simplify the left side.

Simplify the expression.

Move the negative one from the denominator of $\frac{- x + 1}{- 1}$ .

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

Rewrite $- 1 \cdot (- x + 1)$ as $- (- x + 1)$ .

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

Apply the distributive property.

$- - x - 1 \cdot 1 = \frac{3}{- 2}$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$1 x - 1 \cdot 1 = \frac{3}{- 2}$

Multiply $x$ by $1$ .

$x - 1 \cdot 1 = \frac{3}{- 2}$

Multiply $- 1$ by $1$ .

$x - 1 = \frac{3}{- 2}$

Simplify the right side.

Move the negative in front of the fraction.

$x - 1 = - \frac{3}{2}$

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

Add $1$ to both sides of the equation.

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $- 3$ and $2$ .

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

Move the negative in front of the fraction.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.5$

Solve for x 216=(1/36)^(x-1)

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