Simplify (4z-4/z)/(8+4/z)

$\frac{4 z - \frac{4}{z}}{8 + \frac{4}{z}}$

Step 1

Cancel the common factor of $4 z - \frac{4}{z}$ and $8 + \frac{4}{z}$ .

Factor $4$ out of $4 z$ .

$\frac{4 (z) - \frac{4}{z}}{8 + \frac{4}{z}}$

Factor $4$ out of $- \frac{4}{z}$ .

$\frac{4 (z) + 4 (- \frac{1}{z})}{8 + \frac{4}{z}}$

Factor $4$ out of $4 (z) + 4 (- \frac{1}{z})$ .

$\frac{4 (z - \frac{1}{z})}{8 + \frac{4}{z}}$

Cancel the common factors.

Factor $4$ out of $8$ .

$\frac{4 (z - \frac{1}{z})}{4 \cdot 2 + \frac{4}{z}}$

Factor $4$ out of $\frac{4}{z}$ .

$\frac{4 (z - \frac{1}{z})}{4 \cdot 2 + 4 \frac{1}{z}}$

Factor $4$ out of $4 \cdot 2 + 4 \frac{1}{z}$ .

$\frac{4 (z - \frac{1}{z})}{4 \cdot (2 + \frac{1}{z})}$

Cancel the common factor.

$\frac{4 (z - \frac{1}{z})}{4 \cdot (2 + \frac{1}{z})}$

Rewrite the expression.

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

Step 2

Multiply the numerator and denominator of the fraction by $z$ .

Multiply $\frac{z - \frac{1}{z}}{2 + \frac{1}{z}}$ by $\frac{z}{z}$ .

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

Combine.

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

Step 3

Apply the distributive property.

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

Step 4

Simplify by cancelling.

Cancel the common factor of $z$ .

Move the leading negative in $- \frac{1}{z}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $z$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Simplify the numerator.

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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 $b = 1$ .

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

Step 6

Move $2$ to the left of $z$ .

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

Simplify (4z-4/z)/(8+4/z)

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