Finite Math Examples

$\frac{4^{2 + n} \cdot 2^{4 n + 1} \cdot 8^{2 - n}}{2^{3 n + 1}}$

Step 1

Cancel the common factor of $2^{4 n + 1}$ and $2^{3 n + 1}$ .

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Step 1.1

Factor $2^{3 n + 1}$ out of $4^{2 + n} \cdot 2^{4 n + 1} \cdot 8^{2 - n}$ .

$\frac{2^{3 n + 1} (4^{2 + n} \cdot 2^{4 n + 1 - (3 n + 1)} \cdot 8^{2 - n})}{2^{3 n + 1}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Multiply by $1$ .

$\frac{2^{3 n + 1} (4^{2 + n} \cdot 2^{4 n + 1 - (3 n + 1)} \cdot 8^{2 - n})}{2^{3 n + 1} \cdot 1}$

Step 1.2.2

Cancel the common factor.

$\frac{2^{3 n + 1} (4^{2 + n} \cdot 2^{4 n + 1 - (3 n + 1)} \cdot 8^{2 - n})}{2^{3 n + 1} \cdot 1}$

Step 1.2.3

Rewrite the expression.

$\frac{4^{2 + n} \cdot 2^{4 n + 1 - (3 n + 1)} \cdot 8^{2 - n}}{1}$

Step 1.2.4

Divide $4^{2 + n} \cdot 2^{4 n + 1 - (3 n + 1)} \cdot 8^{2 - n}$ by $1$ .

$4^{2 + n} \cdot 2^{4 n + 1 - (3 n + 1)} \cdot 8^{2 - n}$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$4^{2 + n} \cdot 2^{4 n + 1 - (3 n) - 1 \cdot 1} \cdot 8^{2 - n}$

Step 2.2

Multiply $3$ by $- 1$ .

$4^{2 + n} \cdot 2^{4 n + 1 - 3 n - 1 \cdot 1} \cdot 8^{2 - n}$

Step 2.3

Multiply $- 1$ by $1$ .

$4^{2 + n} \cdot 2^{4 n + 1 - 3 n - 1} \cdot 8^{2 - n}$

Step 3

Simplify by adding terms.

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Step 3.1

Combine the opposite terms in $4 n + 1 - 3 n - 1$ .

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Step 3.1.1

Subtract $1$ from $1$ .

$4^{2 + n} \cdot 2^{4 n - 3 n + 0} \cdot 8^{2 - n}$

Step 3.1.2

Add $4 n - 3 n$ and $0$ .

$4^{2 + n} \cdot 2^{4 n - 3 n} \cdot 8^{2 - n}$

Step 3.2

Subtract $3 n$ from $4 n$ .

$4^{2 + n} \cdot 2^{n} \cdot 8^{2 - n}$

Step 4

Multiply $4^{2 + n} \cdot 2^{n}$ .

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Step 4.1

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

${(2^{2})}^{2 + n} \cdot 2^{n} \cdot 8^{2 - n}$

Step 4.2

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

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Step 4.2.1

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

$2^{2 (2 + n)} \cdot 2^{n} \cdot 8^{2 - n}$

Step 4.2.2

Apply the distributive property.

$2^{2 \cdot 2 + 2 n} \cdot 2^{n} \cdot 8^{2 - n}$

Step 4.2.3

Multiply $2$ by $2$ .

$2^{4 + 2 n} \cdot 2^{n} \cdot 8^{2 - n}$

Step 4.3

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

$2^{4 + 2 n + n} \cdot 8^{2 - n}$

Step 4.4

Add $2 n$ and $n$ .

$2^{4 + 3 n} \cdot 8^{2 - n}$

Step 5

Multiply $2^{4 + 3 n} \cdot 8^{2 - n}$ .

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Step 5.1

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

$2^{4 + 3 n} \cdot {(2^{3})}^{2 - n}$

Step 5.2

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

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Step 5.2.1

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

$2^{4 + 3 n} \cdot 2^{3 (2 - n)}$

Step 5.2.2

Apply the distributive property.

$2^{4 + 3 n} \cdot 2^{3 \cdot 2 + 3 (- n)}$

Step 5.2.3

Multiply $3$ by $2$ .

$2^{4 + 3 n} \cdot 2^{6 + 3 (- n)}$

Step 5.2.4

Multiply $- 1$ by $3$ .

$2^{4 + 3 n} \cdot 2^{6 - 3 n}$

Step 5.3

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

$2^{4 + 3 n + 6 - 3 n}$

Step 5.4

Add $4$ and $6$ .

$2^{3 n + 10 - 3 n}$

Step 5.5

Subtract $3 n$ from $3 n$ .

$2^{0 + 10}$

Step 5.6

Add $0$ and $10$ .

$2^{10}$

Step 6

Raise $2$ to the power of $10$ .

$1024$