Solve for n (1/36)^n=216

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

Step 1

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

$\frac{1^{n}}{36^{n}} = 216$

Step 2

One to any power is one.

$\frac{1}{36^{n}} = 216$

Step 3

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

$36^{- n} = 216$

Step 4

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

$6^{2 (- n)} = 6^{3}$

Step 5

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

$2 (- n) = 3$

Step 6

Solve for $n$ .

Multiply $- 1$ by $2$ .

$- 2 n = 3$

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

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

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $n$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$n = - 1.5$

Mixed Number Form:

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