Finite Math Examples

$\frac{2^{n + 1}}{{(2^{n})}^{n - 1}} \cdot \frac{{(2^{n - 1})}^{n + 1}}{4^{n + 1}}$

Step 1

Combine.

$\frac{2^{n + 1} {(2^{n - 1})}^{n + 1}}{{(2^{n})}^{n - 1} \cdot 4^{n + 1}}$

Step 2

Apply basic rules of exponents.

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Step 2.1

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

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Step 2.1.1

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

$\frac{2^{n + 1} \cdot 2^{(n - 1) (n + 1)}}{{(2^{n})}^{n - 1} \cdot 4^{n + 1}}$

Step 2.1.2

Expand $(n - 1) (n + 1)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$\frac{2^{n + 1} \cdot 2^{n (n + 1) - 1 (n + 1)}}{{(2^{n})}^{n - 1} \cdot 4^{n + 1}}$

Step 2.1.2.2

Apply the distributive property.

$\frac{2^{n + 1} \cdot 2^{n \cdot n + n \cdot 1 - 1 (n + 1)}}{{(2^{n})}^{n - 1} \cdot 4^{n + 1}}$

Step 2.1.2.3

Apply the distributive property.

$\frac{2^{n + 1} \cdot 2^{n \cdot n + n \cdot 1 - 1 n - 1 \cdot 1}}{{(2^{n})}^{n - 1} \cdot 4^{n + 1}}$

Step 2.1.3

Combine the opposite terms in $n \cdot n + n \cdot 1 - 1 n - 1 \cdot 1$ .

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Step 2.1.3.1

Reorder the factors in the terms $n \cdot 1$ and $- 1 n$ .

$\frac{2^{n + 1} \cdot 2^{n \cdot n + 1 n - 1 n - 1 \cdot 1}}{{(2^{n})}^{n - 1} \cdot 4^{n + 1}}$

Step 2.1.3.2

Subtract $1 n$ from $1 n$ .

$\frac{2^{n + 1} \cdot 2^{n \cdot n + 0 - 1 \cdot 1}}{{(2^{n})}^{n - 1} \cdot 4^{n + 1}}$

Step 2.1.3.3

Add $n \cdot n$ and $0$ .

$\frac{2^{n + 1} \cdot 2^{n \cdot n - 1 \cdot 1}}{{(2^{n})}^{n - 1} \cdot 4^{n + 1}}$

Step 2.1.4

Simplify each term.

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Step 2.1.4.1

Multiply $n$ by $n$ .

$\frac{2^{n + 1} \cdot 2^{n^{2} - 1 \cdot 1}}{{(2^{n})}^{n - 1} \cdot 4^{n + 1}}$

Step 2.1.4.2

Multiply $- 1$ by $1$ .

$\frac{2^{n + 1} \cdot 2^{n^{2} - 1}}{{(2^{n})}^{n - 1} \cdot 4^{n + 1}}$

Step 2.2

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

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Step 2.2.1

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

$\frac{2^{n + 1} \cdot 2^{n^{2} - 1}}{2^{n (n - 1)} \cdot 4^{n + 1}}$

Step 2.2.2

Apply the distributive property.

$\frac{2^{n + 1} \cdot 2^{n^{2} - 1}}{2^{n \cdot n + n \cdot - 1} \cdot 4^{n + 1}}$

Step 2.2.3

Multiply $n$ by $n$ .

$\frac{2^{n + 1} \cdot 2^{n^{2} - 1}}{2^{n^{2} + n \cdot - 1} \cdot 4^{n + 1}}$

Step 2.2.4

Move $- 1$ to the left of $n$ .

$\frac{2^{n + 1} \cdot 2^{n^{2} - 1}}{2^{n^{2} - 1 \cdot n} \cdot 4^{n + 1}}$

Step 2.2.5

Rewrite $- 1 n$ as $- n$ .

$\frac{2^{n + 1} \cdot 2^{n^{2} - 1}}{2^{n^{2} - n} \cdot 4^{n + 1}}$

Step 3

Multiply $2^{n + 1}$ by $2^{n^{2} - 1}$ by adding the exponents.

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Step 3.1

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

$\frac{2^{n + 1 + n^{2} - 1}}{2^{n^{2} - n} \cdot 4^{n + 1}}$

Step 3.2

Combine the opposite terms in $n + 1 + n^{2} - 1$ .

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Step 3.2.1

Subtract $1$ from $1$ .

$\frac{2^{n + n^{2} + 0}}{2^{n^{2} - n} \cdot 4^{n + 1}}$

Step 3.2.2

Add $n + n^{2}$ and $0$ .

$\frac{2^{n + n^{2}}}{2^{n^{2} - n} \cdot 4^{n + 1}}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $4$ as $2^{2}$ .

$\frac{2^{n + n^{2}}}{2^{n^{2} - n} \cdot {(2^{2})}^{n + 1}}$

Step 4.2

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

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Step 4.2.1

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

$\frac{2^{n + n^{2}}}{2^{n^{2} - n} \cdot 2^{2 (n + 1)}}$

Step 4.2.2

Apply the distributive property.

$\frac{2^{n + n^{2}}}{2^{n^{2} - n} \cdot 2^{2 n + 2 \cdot 1}}$

Step 4.2.3

Multiply $2$ by $1$ .

$\frac{2^{n + n^{2}}}{2^{n^{2} - n} \cdot 2^{2 n + 2}}$

Step 4.3

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

$\frac{2^{n + n^{2}}}{2^{n^{2} - n + 2 n + 2}}$

Step 4.4

Add $- n$ and $2 n$ .

$\frac{2^{n + n^{2}}}{2^{n^{2} + n + 2}}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $2^{n + n^{2}}$ and $2^{n^{2} + n + 2}$ .

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Step 5.1.1

Factor $2^{n^{2} + n + 2}$ out of $2^{n + n^{2}}$ .

$\frac{2^{n^{2} + n + 2} \cdot 2^{n + n^{2} - (n^{2} + n + 2)}}{2^{n^{2} + n + 2}}$

Step 5.1.2

Cancel the common factors.

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Step 5.1.2.1

Multiply by $1$ .

$\frac{2^{n^{2} + n + 2} \cdot 2^{n + n^{2} - (n^{2} + n + 2)}}{2^{n^{2} + n + 2} \cdot 1}$

Step 5.1.2.2

Cancel the common factor.

$\frac{2^{n^{2} + n + 2} \cdot 2^{n + n^{2} - (n^{2} + n + 2)}}{2^{n^{2} + n + 2} \cdot 1}$

Step 5.1.2.3

Rewrite the expression.

$\frac{2^{n + n^{2} - (n^{2} + n + 2)}}{1}$

Step 5.1.2.4

Divide $2^{n + n^{2} - (n^{2} + n + 2)}$ by $1$ .

$2^{n + n^{2} - (n^{2} + n + 2)}$

Step 5.2

Simplify each term.

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Step 5.2.1

Apply the distributive property.

$2^{n + n^{2} - n^{2} - n - 1 \cdot 2}$

Step 5.2.2

Multiply $- 1$ by $2$ .

$2^{n + n^{2} - n^{2} - n - 2}$

Step 5.3

Simplify by adding terms.

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Step 5.3.1

Combine the opposite terms in $n + n^{2} - n^{2} - n - 2$ .

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Step 5.3.1.1

Subtract $n^{2}$ from $n^{2}$ .

$2^{n + 0 - n - 2}$

Step 5.3.1.2

Add $n$ and $0$ .

$2^{n - n - 2}$

Step 5.3.1.3

Subtract $n$ from $n$ .

$2^{0 - 2}$

Step 5.3.2

Subtract $2$ from $0$ .

$2^{- 2}$

Step 6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2^{2}}$

Step 7

Raise $2$ to the power of $2$ .

$\frac{1}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{4}$

Decimal Form:

$0.25$