Basic Math Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Combine.

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

Step 3

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

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Step 3.1

Multiply $2$ by $2^{n + 1}$ .

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Step 3.1.1

Raise $2$ to the power of $1$ .

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

Step 3.1.2

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

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

Step 3.2

Add $1$ and $1$ .

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

Step 4

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

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Step 4.1

Move $2^{n + 2}$ .

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

Step 4.2

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

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

Step 4.3

Subtract $1$ from $2$ .

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

Step 5

Cancel the common factor of $2^{n + 1}$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

Apply basic rules of exponents.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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

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Step 6.1.2.1

Apply the distributive property.

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

Step 6.1.2.2

Apply the distributive property.

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

Step 6.1.2.3

Apply the distributive property.

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

Step 6.1.3

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

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Step 6.1.3.1

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

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

Step 6.1.3.2

Subtract $1 n$ from $1 n$ .

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

Step 6.1.3.3

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

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

Step 6.1.4

Simplify each term.

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Step 6.1.4.1

Multiply $n$ by $n$ .

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

Step 6.1.4.2

Multiply $- 1$ by $1$ .

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Multiply $n$ by $n$ .

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

Step 7

Cancel the common factor of $2^{n^{2} - 1}$ and $2^{n^{2}}$ .

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Step 7.1

Factor $2^{n^{2}}$ out of $2^{n^{2} - 1}$ .

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

Multiply by $1$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

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

Step 7.2.4

Divide $2^{- 1}$ by $1$ .

$2^{- 1}$

Step 8

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

$\frac{1}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$