Basic Math Examples

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

Step 1

Multiply $3$ by $3^{q + 4}$ by adding the exponents.

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Step 1.1

Multiply $3$ by $3^{q + 4}$ .

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.2

Add $1$ and $4$ .

$\frac{3^{q + 3} - 3^{2} \cdot 3^{q}}{3^{q + 5}}$

Step 2

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

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Step 2.1

Move $3^{q}$ .

$\frac{3^{q + 3} - (3^{q} \cdot 3^{2})}{3^{q + 5}}$

Step 2.2

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

$\frac{3^{q + 3} - 3^{q + 2}}{3^{q + 5}}$

Step 3

Split the fraction $\frac{3^{q + 3} - 3^{q + 2}}{3^{q + 5}}$ into two fractions.

$\frac{3^{q + 3}}{3^{q + 5}} + \frac{- 3^{q + 2}}{3^{q + 5}}$

Step 4

Cancel the common factor of $3^{q + 3}$ and $3^{q + 5}$ .

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Step 4.1

Factor $3^{q + 5}$ out of $3^{q + 3}$ .

$\frac{3^{q + 5} \cdot 3^{q + 3 - (q + 5)}}{3^{q + 5}} + \frac{- 3^{q + 2}}{3^{q + 5}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Multiply by $1$ .

$\frac{3^{q + 5} \cdot 3^{q + 3 - (q + 5)}}{3^{q + 5} \cdot 1} + \frac{- 3^{q + 2}}{3^{q + 5}}$

Step 4.2.2

Cancel the common factor.

$\frac{3^{q + 5} \cdot 3^{q + 3 - (q + 5)}}{3^{q + 5} \cdot 1} + \frac{- 3^{q + 2}}{3^{q + 5}}$

Step 4.2.3

Rewrite the expression.

$\frac{3^{q + 3 - (q + 5)}}{1} + \frac{- 3^{q + 2}}{3^{q + 5}}$

Step 4.2.4

Divide $3^{q + 3 - (q + 5)}$ by $1$ .

$3^{q + 3 - (q + 5)} + \frac{- 3^{q + 2}}{3^{q + 5}}$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

$3^{q + 3 - q - 1 \cdot 5} + \frac{- 3^{q + 2}}{3^{q + 5}}$

Step 5.2

Multiply $- 1$ by $5$ .

$3^{q + 3 - q - 5} + \frac{- 3^{q + 2}}{3^{q + 5}}$

Step 6

Simplify by adding terms.

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Step 6.1

Combine the opposite terms in $q + 3 - q - 5$ .

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Step 6.1.1

Subtract $q$ from $q$ .

$3^{0 + 3 - 5} + \frac{- 3^{q + 2}}{3^{q + 5}}$

Step 6.1.2

Add $0$ and $3$ .

$3^{3 - 5} + \frac{- 3^{q + 2}}{3^{q + 5}}$

Step 6.2

Subtract $5$ from $3$ .

$3^{- 2} + \frac{- 3^{q + 2}}{3^{q + 5}}$

Step 7

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

$\frac{1}{3^{2}} + \frac{- 3^{q + 2}}{3^{q + 5}}$

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

Raise $3$ to the power of $2$ .

$\frac{1}{9} + \frac{- 3^{q + 2}}{3^{q + 5}}$

Step 8.2

Cancel the common factor of $3^{q + 2}$ and $3^{q + 5}$ .

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Step 8.2.1

Factor $3^{q + 5}$ out of $- 3^{q + 2}$ .

$\frac{1}{9} + \frac{3^{q + 5} (- 3^{q + 2 - (q + 5)})}{3^{q + 5}}$

Step 8.2.2

Cancel the common factors.

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Step 8.2.2.1

Multiply by $1$ .

$\frac{1}{9} + \frac{3^{q + 5} (- 3^{q + 2 - (q + 5)})}{3^{q + 5} \cdot 1}$

Step 8.2.2.2

Cancel the common factor.

$\frac{1}{9} + \frac{3^{q + 5} (- 3^{q + 2 - (q + 5)})}{3^{q + 5} \cdot 1}$

Step 8.2.2.3

Rewrite the expression.

$\frac{1}{9} + \frac{- 3^{q + 2 - (q + 5)}}{1}$

Step 8.2.2.4

Divide $- 3^{q + 2 - (q + 5)}$ by $1$ .

$\frac{1}{9} - 3^{q + 2 - (q + 5)}$

Step 9

Simplify each term.

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Step 9.1

Apply the distributive property.

$\frac{1}{9} - 3^{q + 2 - q - 1 \cdot 5}$

Step 9.2

Multiply $- 1$ by $5$ .

$\frac{1}{9} - 3^{q + 2 - q - 5}$

Step 10

Simplify by adding terms.

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Step 10.1

Combine the opposite terms in $q + 2 - q - 5$ .

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Step 10.1.1

Subtract $q$ from $q$ .

$\frac{1}{9} - 3^{0 + 2 - 5}$

Step 10.1.2

Add $0$ and $2$ .

$\frac{1}{9} - 3^{2 - 5}$

Step 10.2

Subtract $5$ from $2$ .

$\frac{1}{9} - 3^{- 3}$

Step 11

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

$\frac{1}{9} - \frac{1}{3^{3}}$

Step 12

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

$\frac{1}{9} - \frac{1}{27}$

Step 13

To write $\frac{1}{9}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{9} \cdot \frac{3}{3} - \frac{1}{27}$

Step 14

Write each expression with a common denominator of $27$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.1

Multiply $\frac{1}{9}$ by $\frac{3}{3}$ .

$\frac{3}{9 \cdot 3} - \frac{1}{27}$

Step 14.2

Multiply $9$ by $3$ .

$\frac{3}{27} - \frac{1}{27}$

Step 15

Combine the numerators over the common denominator.

$\frac{3 - 1}{27}$

Step 16

Subtract $1$ from $3$ .

$\frac{2}{27}$

Step 17

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{27}$

Decimal Form:

$0.\overline{074}$