Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

$2 n = - 3$

Step 5

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

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Divide $n$ by $1$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$n = - 1.5$

Mixed Number Form:

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