Algebra Examples

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

Step 1

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

${(6^{2})}^{n - 3} \cdot 216^{n} = 216^{2 n + 1}$

Step 2

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

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Step 2.1

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

$6^{2 (n - 3)} \cdot 216^{n} = 216^{2 n + 1}$

Step 2.2

Apply the distributive property.

$6^{2 n + 2 \cdot - 3} \cdot 216^{n} = 216^{2 n + 1}$

Step 2.3

Multiply $2$ by $- 3$ .

$6^{2 n - 6} \cdot 216^{n} = 216^{2 n + 1}$

Step 3

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

$6^{2 n - 6} \cdot {(6^{3})}^{n} = 216^{2 n + 1}$

Step 4

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

$6^{2 n - 6} \cdot 6^{3 n} = 216^{2 n + 1}$

Step 5

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

$6^{2 n - 6 + 3 n} = 216^{2 n + 1}$

Step 6

Add $2 n$ and $3 n$ .

$6^{5 n - 6} = 216^{2 n + 1}$

Step 7

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

$6^{5 n - 6} = 6^{3 (2 n + 1)}$

Step 8

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

$5 n - 6 = 3 (2 n + 1)$

Step 9

Solve for $n$ .

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Step 9.1

Simplify $3 (2 n + 1)$ .

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Step 9.1.1

Rewrite.

$5 n - 6 = 0 + 0 + 3 (2 n + 1)$

Step 9.1.2

Simplify by adding zeros.

$5 n - 6 = 3 (2 n + 1)$

Step 9.1.3

Apply the distributive property.

$5 n - 6 = 3 (2 n) + 3 \cdot 1$

Step 9.1.4

Multiply.

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Step 9.1.4.1

Multiply $2$ by $3$ .

$5 n - 6 = 6 n + 3 \cdot 1$

Step 9.1.4.2

Multiply $3$ by $1$ .

$5 n - 6 = 6 n + 3$

Step 9.2

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

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Step 9.2.1

Subtract $6 n$ from both sides of the equation.

$5 n - 6 - 6 n = 3$

Step 9.2.2

Subtract $6 n$ from $5 n$ .

$- n - 6 = 3$

Step 9.3

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

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Step 9.3.1

Add $6$ to both sides of the equation.

$- n = 3 + 6$

Step 9.3.2

Add $3$ and $6$ .

$- n = 9$

Step 9.4

Divide each term in $- n = 9$ by $- 1$ and simplify.

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Step 9.4.1

Divide each term in $- n = 9$ by $- 1$ .

$\frac{- n}{- 1} = \frac{9}{- 1}$

Step 9.4.2

Simplify the left side.

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Step 9.4.2.1

Dividing two negative values results in a positive value.

$\frac{n}{1} = \frac{9}{- 1}$

Step 9.4.2.2

Divide $n$ by $1$ .

$n = \frac{9}{- 1}$

Step 9.4.3

Simplify the right side.

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Step 9.4.3.1

Divide $9$ by $- 1$ .

$n = - 9$