Solve for n (216^(n-2))/((1/36)^(3n))=216

$\frac{216^{n - 2}}{{(\frac{1}{36})}^{3 n}} = 216$

Step 1

Multiply both sides by ${(\frac{1}{36})}^{3 n}$ .

$\frac{216^{n - 2}}{{(\frac{1}{36})}^{3 n}} {(\frac{1}{36})}^{3 n} = 216 {(\frac{1}{36})}^{3 n}$

Step 2

Simplify.

Simplify the left side.

Cancel the common factor of ${(\frac{1}{36})}^{3 n}$ .

Cancel the common factor.

$\frac{216^{n - 2}}{{(\frac{1}{36})}^{3 n}} {(\frac{1}{36})}^{3 n} = 216 {(\frac{1}{36})}^{3 n}$

Rewrite the expression.

$216^{n - 2} = 216 {(\frac{1}{36})}^{3 n}$

Simplify the right side.

Simplify $216 {(\frac{1}{36})}^{3 n}$ .

Simplify the expression.

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

$216^{n - 2} = 216 \frac{1^{3 n}}{36^{3 n}}$

One to any power is one.

$216^{n - 2} = 216 \frac{1}{36^{3 n}}$

Combine $216$ and $\frac{1}{36^{3 n}}$ .

$216^{n - 2} = \frac{216}{36^{3 n}}$

Step 3

Solve for $n$ .

Move $36^{3 n}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$216^{n - 2} = 216 \cdot 36^{- 3 n}$

Rewrite $216$ as $6^{3}$ .

$216^{n - 2} = 6^{3} \cdot 36^{- 3 n}$

Rewrite $36$ as $6^{2}$ .

$216^{n - 2} = 6^{3} \cdot {(6^{2})}^{- 3 n}$

Multiply the exponents in ${(6^{2})}^{- 3 n}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$216^{n - 2} = 6^{3} \cdot 6^{2 (- 3 n)}$

Multiply $- 3$ by $2$ .

$216^{n - 2} = 6^{3} \cdot 6^{- 6 n}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$216^{n - 2} = 6^{3 - 6 n}$

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

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

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

$3 (n - 2) = 3 - 6 n$

Solve for $n$ .

Simplify $3 (n - 2)$ .

Rewrite.

$0 + 0 + 3 (n - 2) = 3 - 6 n$

Simplify by adding zeros.

$3 (n - 2) = 3 - 6 n$

Apply the distributive property.

$3 n + 3 \cdot - 2 = 3 - 6 n$

Multiply $3$ by $- 2$ .

$3 n - 6 = 3 - 6 n$

Move all terms containing $n$ to the left side of the equation.

Add $6 n$ to both sides of the equation.

$3 n - 6 + 6 n = 3$

Add $3 n$ and $6 n$ .

$9 n - 6 = 3$

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

Add $6$ to both sides of the equation.

$9 n = 3 + 6$

Add $3$ and $6$ .

$9 n = 9$

Divide each term in $9 n = 9$ by $9$ and simplify.

Divide each term in $9 n = 9$ by $9$ .

$\frac{9 n}{9} = \frac{9}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 n}{9} = \frac{9}{9}$

Divide $n$ by $1$ .

$n = \frac{9}{9}$

Simplify the right side.

Divide $9$ by $9$ .

$n = 1$

Solve for n (216^(n-2))/((1/36)^(3n))=216

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