Basic Math Examples

$\frac{9^{n}}{3^{n (n + 1)}} \div \frac{3^{n + 1}}{3^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Combine.

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

Step 3

Multiply $3^{n (n + 1)}$ by $3^{n + 1}$ 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.

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

Step 3.2

Simplify each term.

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

Multiply $n$ by $n$ .

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

Step 3.2.3

Multiply $n$ by $1$ .

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

Step 3.3

Add $n$ and $n$ .

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

Step 4

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

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Step 4.1

Factor $3^{n^{2} + 2 n + 1}$ out of $9^{n} \cdot 3^{2}$ .

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Multiply by $1$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 4.2.4

Divide $9^{n} \cdot 3^{2 - (n^{2} + 2 n + 1)}$ by $1$ .

$9^{n} \cdot 3^{2 - (n^{2} + 2 n + 1)}$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

$9^{n} \cdot 3^{2 - n^{2} - (2 n) - 1 \cdot 1}$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $2$ by $- 1$ .

$9^{n} \cdot 3^{2 - n^{2} - 2 n - 1 \cdot 1}$

Step 5.2.2

Multiply $- 1$ by $1$ .

$9^{n} \cdot 3^{2 - n^{2} - 2 n - 1}$

Step 6

Subtract $1$ from $2$ .

$9^{n} \cdot 3^{- n^{2} - 2 n + 1}$

Step 7

Multiply $9^{n} \cdot 3^{- n^{2} - 2 n + 1}$ .

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

$3^{2 n - n^{2} - 2 n + 1}$

Step 7.4

Subtract $2 n$ from $2 n$ .

$3^{- n^{2} + 0 + 1}$

Step 7.5

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

$3^{- n^{2} + 1}$