Finite Math Examples

$a (n) = 32 \cdot {(\frac{1}{8})}^{n - 1}$

Step 1

Simplify.

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Step 1.1

Rewrite $32$ as $2^{5}$ .

$a n = 2^{5} \cdot {(\frac{1}{8})}^{n - 1}$

Step 1.2

Rewrite $\frac{1}{8}$ as $8^{- 1}$ .

$a n = 2^{5} \cdot {(8^{- 1})}^{n - 1}$

Step 1.3

Rewrite $8$ as $2^{3}$ .

$a n = 2^{5} \cdot {({(2^{3})}^{- 1})}^{n - 1}$

Step 1.4

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

$a n = 2^{5} \cdot {(2^{3 \cdot - 1})}^{n - 1}$

Step 1.5

Multiply $3$ by $- 1$ .

$a n = 2^{5} \cdot {(2^{- 3})}^{n - 1}$

Step 1.6

Multiply the exponents in ${(2^{- 3})}^{n - 1}$ .

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Step 1.6.1

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

$a n = 2^{5} \cdot 2^{- 3 (n - 1)}$

Step 1.6.2

Apply the distributive property.

$a n = 2^{5} \cdot 2^{- 3 n - 3 \cdot - 1}$

Step 1.6.3

Multiply $- 3$ by $- 1$ .

$a n = 2^{5} \cdot 2^{- 3 n + 3}$

Step 1.7

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

$a n = 2^{5 - 3 n + 3}$

Step 1.8

Add $5$ and $3$ .

$a n = 2^{- 3 n + 8}$

Step 2

Divide each term in $a n = 2^{- 3 n + 8}$ by $n$ and simplify.

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Step 2.1

Divide each term in $a n = 2^{- 3 n + 8}$ by $n$ .

$\frac{a n}{n} = \frac{2^{- 3 n + 8}}{n}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $n$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{a n}{n} = \frac{2^{- 3 n + 8}}{n}$

Step 2.2.1.2

Divide $a$ by $1$ .

$a = \frac{2^{- 3 n + 8}}{n}$