Basic Math Examples

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

Step 1

Combine.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

Subtract $1$ from $1$ .

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

Step 2.2.2

Add $n + n$ and $0$ .

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

Step 2.3

Add $n$ and $n$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $2^{2 n}$ out of $2^{2 n} \cdot 4^{- n}$ .

$a = \frac{2^{2 n} (2^{- n} {(2^{n - 1})}^{n})}{2^{2 n} (4^{- n})}$

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

$a = \frac{2^{- n} {(2^{n - 1})}^{n}}{4^{- n}}$

Step 5

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

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Step 5.1

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

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply $n$ by $n$ .

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

Step 5.4

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

$a = \frac{2^{- n} \cdot 2^{n^{2} - n}}{4^{- n}}$

Step 6

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

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Step 6.1

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

$a = \frac{2^{- n + n^{2} - n}}{4^{- n}}$

Step 6.2

Subtract $n$ from $- n$ .

$a = \frac{2^{n^{2} - 2 n}}{4^{- n}}$