Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Multiply $3$ by $3^{2 - n}$ by adding the exponents.

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Step 1.1.1

Multiply $3$ by $3^{2 - n}$ .

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Step 1.1.1.1

Raise $3$ to the power of $1$ .

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

Step 1.1.1.2

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

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

Step 1.1.2

Add $1$ and $2$ .

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

Step 1.2

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

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Step 1.2.1

Factor out negative.

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Add $2$ and $1$ .

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

Step 1.3

Subtract $3^{3 - n}$ from $3^{3 - n}$ .

$\frac{3^{- 3 - n} + 0}{9 \cdot 3^{2 - n}}$

Step 1.4

Add $3^{- 3 - n}$ and $0$ .

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

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{3^{- 3 - n}}{3^{2} \cdot 3^{2 - n}}$

Step 2.2

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

$\frac{3^{- 3 - n}}{3^{2 + 2 - n}}$

Step 2.3

Add $2$ and $2$ .

$\frac{3^{- 3 - n}}{3^{4 - n}}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $3^{- 3 - n}$ and $3^{4 - n}$ .

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Step 3.1.1

Factor $3^{4 - n}$ out of $3^{- 3 - n}$ .

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Multiply by $1$ .

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

$\frac{3^{- 3 - n - (4 - n)}}{1}$

Step 3.1.2.4

Divide $3^{- 3 - n - (4 - n)}$ by $1$ .

$3^{- 3 - n - (4 - n)}$

Step 3.2

Simplify each term.

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Step 3.2.1

Apply the distributive property.

$3^{- 3 - n - 1 \cdot 4 - - n}$

Step 3.2.2

Multiply $- 1$ by $4$ .

$3^{- 3 - n - 4 - - n}$

Step 3.2.3

Multiply $- - n$ .

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Step 3.2.3.1

Multiply $- 1$ by $- 1$ .

$3^{- 3 - n - 4 + 1 n}$

Step 3.2.3.2

Multiply $n$ by $1$ .

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

Step 3.3

Simplify by adding terms.

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Step 3.3.1

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

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Step 3.3.1.1

Add $- n$ and $n$ .

$3^{- 3 + 0 - 4}$

Step 3.3.1.2

Add $- 3$ and $0$ .

$3^{- 3 - 4}$

Step 3.3.2

Subtract $4$ from $- 3$ .

$3^{- 7}$

Step 4

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

$\frac{1}{3^{7}}$

Step 5

Raise $3$ to the power of $7$ .

$\frac{1}{2187}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2187}$

Decimal Form:

$0.00045724 \dots$