Basic Math Examples

q - \frac{1}{q} \div q + \frac{1}{q}

$q - \frac{1}{q} \div q + \frac{1}{q}$

Step 1

Simplify terms.

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Step 1.1

Combine the numerators over the common denominator.

q + \frac{- \frac{1}{q} + 1}{q}

$q + \frac{- \frac{1}{q} + 1}{q}$

Step 1.2

Write

1

$1$ as a fraction with a common denominator.

q + \frac{- \frac{1}{q} + \frac{q}{q}}{q}

$q + \frac{- \frac{1}{q} + \frac{q}{q}}{q}$

Step 1.3

Combine the numerators over the common denominator.

q + \frac{\frac{- 1 + q}{q}}{q}

$q + \frac{\frac{- 1 + q}{q}}{q}$

Step 1.4

Rewrite

- 1

$- 1$ as

- 1 (1)

$- 1 (1)$ .

q + \frac{\frac{- 1 (1) + q}{q}}{q}

$q + \frac{\frac{- 1 (1) + q}{q}}{q}$

Step 1.5

Factor

- 1

$- 1$ out of

q

$q$ .

q + \frac{\frac{- 1 (1) - 1 (- q)}{q}}{q}

$q + \frac{\frac{- 1 (1) - 1 (- q)}{q}}{q}$

Step 1.6

Factor

- 1

$- 1$ out of

- 1 (1) - 1 (- q)

$- 1 (1) - 1 (- q)$ .

q + \frac{\frac{- 1 (1 - q)}{q}}{q}

$q + \frac{\frac{- 1 (1 - q)}{q}}{q}$

Step 1.7

Move the negative in front of the fraction.

q + \frac{- \frac{1 - q}{q}}{q}

$q + \frac{- \frac{1 - q}{q}}{q}$

q + \frac{- \frac{1 - q}{q}}{q}

$q + \frac{- \frac{1 - q}{q}}{q}$

Step 2

Simplify each term.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

q - \frac{1 - q}{q} \cdot \frac{1}{q}

$q - \frac{1 - q}{q} \cdot \frac{1}{q}$

Step 2.2

Multiply

- \frac{1 - q}{q} \cdot \frac{1}{q}

$- \frac{1 - q}{q} \cdot \frac{1}{q}$ .

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Step 2.2.1

Multiply

\frac{1}{q}

$\frac{1}{q}$ by

\frac{1 - q}{q}

$\frac{1 - q}{q}$ .

q - \frac{1 - q}{q \cdot q}

$q - \frac{1 - q}{q \cdot q}$

Step 2.2.2

Raise

q

$q$ to the power of

1

$1$ .

q - \frac{1 - q}{q^{1} q}

$q - \frac{1 - q}{q^{1} q}$

Step 2.2.3

Raise

q

$q$ to the power of

1

$1$ .

q - \frac{1 - q}{q^{1} q^{1}}

$q - \frac{1 - q}{q^{1} q^{1}}$

Step 2.2.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

q - \frac{1 - q}{q^{1 + 1}}

$q - \frac{1 - q}{q^{1 + 1}}$

Step 2.2.5

Add

1

$1$ and

1

$1$ .

q - \frac{1 - q}{q^{2}}

$q - \frac{1 - q}{q^{2}}$

q - \frac{1 - q}{q^{2}}

$q - \frac{1 - q}{q^{2}}$

q - \frac{1 - q}{q^{2}}

$q - \frac{1 - q}{q^{2}}$

Step 3

To write

q

$q$ as a fraction with a common denominator, multiply by

\frac{q^{2}}{q^{2}}

$\frac{q^{2}}{q^{2}}$ .

\frac{q \cdot q^{2}}{q^{2}} - \frac{1 - q}{q^{2}}

$\frac{q \cdot q^{2}}{q^{2}} - \frac{1 - q}{q^{2}}$

Step 4

Combine the numerators over the common denominator.

\frac{q \cdot q^{2} - (1 - q)}{q^{2}}

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply

q

$q$ by

q^{2}

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

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Step 5.1.1

Multiply

q

$q$ by

q^{2}

$q^{2}$ .

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Step 5.1.1.1

Raise

q

$q$ to the power of

1

$1$ .

\frac{q^{1} q^{2} - (1 - q)}{q^{2}}

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

Step 5.1.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

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

Step 5.1.2

Add

1

$1$ and

2

$2$ .

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

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

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

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

Step 5.2

Apply the distributive property.

\frac{q^{3} - 1 \cdot 1 - - q}{q^{2}}

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

Step 5.3

Multiply

- 1

$- 1$ by

1

$1$ .

\frac{q^{3} - 1 - - q}{q^{2}}

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

Step 5.4

Multiply

- - q

$- - q$ .

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Step 5.4.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

Step 5.4.2

Multiply

q

$q$ by

1

$1$ .

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

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

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

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

Step 5.5

Reorder terms.

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

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

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

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