Basic Math Examples

$\frac{27^{n + 3} \cdot 9^{n + 4}}{3^{n + 1} \cdot 81^{n + 2}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $27$ as $3^{3}$ .

$\frac{{(3^{3})}^{n + 3} \cdot 9^{n + 4}}{3^{n + 1} \cdot 81^{n + 2}}$

Step 1.2

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

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Step 1.2.1

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

$\frac{3^{3 (n + 3)} \cdot 9^{n + 4}}{3^{n + 1} \cdot 81^{n + 2}}$

Step 1.2.2

Apply the distributive property.

$\frac{3^{3 n + 3 \cdot 3} \cdot 9^{n + 4}}{3^{n + 1} \cdot 81^{n + 2}}$

Step 1.2.3

Multiply $3$ by $3$ .

$\frac{3^{3 n + 9} \cdot 9^{n + 4}}{3^{n + 1} \cdot 81^{n + 2}}$

Step 1.3

Rewrite $9$ as $3^{2}$ .

$\frac{3^{3 n + 9} \cdot {(3^{2})}^{n + 4}}{3^{n + 1} \cdot 81^{n + 2}}$

Step 1.4

Multiply the exponents in ${(3^{2})}^{n + 4}$ .

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Step 1.4.1

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

$\frac{3^{3 n + 9} \cdot 3^{2 (n + 4)}}{3^{n + 1} \cdot 81^{n + 2}}$

Step 1.4.2

Apply the distributive property.

$\frac{3^{3 n + 9} \cdot 3^{2 n + 2 \cdot 4}}{3^{n + 1} \cdot 81^{n + 2}}$

Step 1.4.3

Multiply $2$ by $4$ .

$\frac{3^{3 n + 9} \cdot 3^{2 n + 8}}{3^{n + 1} \cdot 81^{n + 2}}$

Step 1.5

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

$\frac{3^{3 n + 9 + 2 n + 8}}{3^{n + 1} \cdot 81^{n + 2}}$

Step 1.6

Add $3 n$ and $2 n$ .

$\frac{3^{5 n + 9 + 8}}{3^{n + 1} \cdot 81^{n + 2}}$

Step 1.7

Add $9$ and $8$ .

$\frac{3^{5 n + 17}}{3^{n + 1} \cdot 81^{n + 2}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $81$ as $3^{4}$ .

$\frac{3^{5 n + 17}}{3^{n + 1} \cdot {(3^{4})}^{n + 2}}$

Step 2.2

Multiply the exponents in ${(3^{4})}^{n + 2}$ .

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Step 2.2.1

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

$\frac{3^{5 n + 17}}{3^{n + 1} \cdot 3^{4 (n + 2)}}$

Step 2.2.2

Apply the distributive property.

$\frac{3^{5 n + 17}}{3^{n + 1} \cdot 3^{4 n + 4 \cdot 2}}$

Step 2.2.3

Multiply $4$ by $2$ .

$\frac{3^{5 n + 17}}{3^{n + 1} \cdot 3^{4 n + 8}}$

Step 2.3

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

$\frac{3^{5 n + 17}}{3^{n + 1 + 4 n + 8}}$

Step 2.4

Add $n$ and $4 n$ .

$\frac{3^{5 n + 17}}{3^{5 n + 1 + 8}}$

Step 2.5

Add $1$ and $8$ .

$\frac{3^{5 n + 17}}{3^{5 n + 9}}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $3^{5 n + 17}$ and $3^{5 n + 9}$ .

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Step 3.1.1

Factor $3^{5 n + 9}$ out of $3^{5 n + 17}$ .

$\frac{3^{5 n + 9} \cdot 3^{5 n + 17 - (5 n + 9)}}{3^{5 n + 9}}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Multiply by $1$ .

$\frac{3^{5 n + 9} \cdot 3^{5 n + 17 - (5 n + 9)}}{3^{5 n + 9} \cdot 1}$

Step 3.1.2.2

Cancel the common factor.

$\frac{3^{5 n + 9} \cdot 3^{5 n + 17 - (5 n + 9)}}{3^{5 n + 9} \cdot 1}$

Step 3.1.2.3

Rewrite the expression.

$\frac{3^{5 n + 17 - (5 n + 9)}}{1}$

Step 3.1.2.4

Divide $3^{5 n + 17 - (5 n + 9)}$ by $1$ .

$3^{5 n + 17 - (5 n + 9)}$

Step 3.2

Simplify each term.

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Step 3.2.1

Apply the distributive property.

$3^{5 n + 17 - (5 n) - 1 \cdot 9}$

Step 3.2.2

Multiply $5$ by $- 1$ .

$3^{5 n + 17 - 5 n - 1 \cdot 9}$

Step 3.2.3

Multiply $- 1$ by $9$ .

$3^{5 n + 17 - 5 n - 9}$

Step 3.3

Simplify by adding terms.

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Step 3.3.1

Combine the opposite terms in $5 n + 17 - 5 n - 9$ .

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Step 3.3.1.1

Subtract $5 n$ from $5 n$ .

$3^{0 + 17 - 9}$

Step 3.3.1.2

Add $0$ and $17$ .

$3^{17 - 9}$

Step 3.3.2

Simplify the expression.

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Step 3.3.2.1

Subtract $9$ from $17$ .

$3^{8}$

Step 3.3.2.2

Raise $3$ to the power of $8$ .

$6561$