Solve for n 3^(2n)=1/27

$3^{2 n} = \frac{1}{27}$

Raise $27$ to the power of $1$ .

$3^{2 n} = \frac{1}{27^{1}}$

Move $27^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$3^{2 n} = 27^{- 1}$

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

$3^{2 n} = 3^{- 3}$

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

$2 n = - 3$

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.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$n = - 1.5$

Mixed Number Form:

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