Trigonometry Examples

${(\frac{1}{36})}^{n} = 216^{n + 5}$

Step 1

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

$\frac{1^{n}}{36^{n}} = 216^{n + 5}$

Step 2

One to any power is one.

$\frac{1}{36^{n}} = 216^{n + 5}$

Step 3

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

$36^{- n} = 216^{n + 5}$

Step 4

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

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

Step 5

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

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

Step 6

Solve for $n$ .

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Simplify $3 (n + 5)$ .

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Step 6.2.1

Apply the distributive property.

$- 2 n = 3 n + 3 \cdot 5$

Step 6.2.2

Multiply $3$ by $5$ .

$- 2 n = 3 n + 15$

Step 6.3

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

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Step 6.3.1

Subtract $3 n$ from both sides of the equation.

$- 2 n - 3 n = 15$

Step 6.3.2

Subtract $3 n$ from $- 2 n$ .

$- 5 n = 15$

Step 6.4

Divide each term in $- 5 n = 15$ by $- 5$ and simplify.

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Step 6.4.1

Divide each term in $- 5 n = 15$ by $- 5$ .

$\frac{- 5 n}{- 5} = \frac{15}{- 5}$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor of $- 5$ .

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Step 6.4.2.1.1

Cancel the common factor.

$\frac{- 5 n}{- 5} = \frac{15}{- 5}$

Step 6.4.2.1.2

Divide $n$ by $1$ .

$n = \frac{15}{- 5}$

Step 6.4.3

Simplify the right side.

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Step 6.4.3.1

Divide $15$ by $- 5$ .

$n = - 3$