Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Factor $q^{- 1}$ out of $q - q^{- 1}$ .

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Step 1.1.1

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

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

Step 1.1.2

Factor $q^{- 1}$ out of $q^{1}$ .

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

Step 1.1.3

Factor $q^{- 1}$ out of $- q^{- 1}$ .

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

Step 1.1.4

Factor $q^{- 1}$ out of $q^{- 1} q^{2} + q^{- 1} \cdot - 1$ .

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

Step 1.2

Rewrite $1$ as $1^{2}$ .

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

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = q$ and $b = 1$ .

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

Step 1.4

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

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

Step 2

Simplify the denominator.

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Step 2.1

Factor $q^{- 1}$ out of $q + q^{- 1}$ .

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Step 2.1.1

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

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

Step 2.1.2

Factor $q^{- 1}$ out of $q^{1}$ .

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

Step 2.1.3

Multiply by $1$ .

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

Step 2.1.4

Factor $q^{- 1}$ out of $q^{- 1} q^{2} + q^{- 1} \cdot 1$ .

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

Step 2.2

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

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

Step 3

Cancel the common factor of $\frac{1}{q}$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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