Finite Math Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.2

Multiply $\frac{z - \frac{1}{z}}{z}$ by $1$ .

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

Step 1.1.3

Simplify the numerator.

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Step 1.1.3.1

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

$1 + \frac{\frac{z \cdot z}{z} - \frac{1}{z}}{z}$

Step 1.1.3.2

Combine the numerators over the common denominator.

$1 + \frac{\frac{z \cdot z - 1}{z}}{z}$

Step 1.1.3.3

Rewrite $\frac{z \cdot z - 1}{z}$ in a factored form.

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Step 1.1.3.3.1

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

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

Step 1.1.3.3.2

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

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

Step 1.1.3.3.3

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

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

Step 1.1.3.3.4

Add $1$ and $1$ .

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

Step 1.1.3.3.5

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

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

Step 1.1.3.3.6

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

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

Step 1.1.4

Multiply the numerator by the reciprocal of the denominator.

$1 + \frac{(z + 1) (z - 1)}{z} \cdot \frac{1}{z}$

Step 1.1.5

Multiply $\frac{(z + 1) (z - 1)}{z} \cdot \frac{1}{z}$ .

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Step 1.1.5.1

Multiply $\frac{(z + 1) (z - 1)}{z}$ by $\frac{1}{z}$ .

$1 + \frac{(z + 1) (z - 1)}{z \cdot z}$

Step 1.1.5.2

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

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

Step 1.1.5.3

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

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

Step 1.1.5.4

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

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

Step 1.1.5.5

Add $1$ and $1$ .

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

Step 1.2

Combine into one fraction.

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Step 1.2.1

Write $1$ as a fraction with a common denominator.

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

Step 1.2.2

Combine the numerators over the common denominator.

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

Step 2

Simplify the numerator.

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Step 2.1

Expand $(z + 1) (z - 1)$ using the FOIL Method.

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Step 2.1.1

Apply the distributive property.

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

Step 2.1.2

Apply the distributive property.

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

Step 2.1.3

Apply the distributive property.

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

Step 2.2

Simplify and combine like terms.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $z$ by $z$ .

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

Step 2.2.1.2

Move $- 1$ to the left of $z$ .

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

Step 2.2.1.3

Rewrite $- 1 z$ as $- z$ .

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

Step 2.2.1.4

Multiply $z$ by $1$ .

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

Step 2.2.1.5

Multiply $- 1$ by $1$ .

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

Step 2.2.2

Add $- z$ and $z$ .

$\frac{z^{2} + z^{2} + 0 - 1}{z^{2}}$

Step 2.2.3

Add $z^{2}$ and $0$ .

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

Step 2.3

Add $z^{2}$ and $z^{2}$ .

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