Simplify (3z-1/(3z))/(1-1/(3z))

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

Step 1

Multiply the numerator and denominator of the fraction by $3 z$ .

Multiply $\frac{3 z - \frac{1}{3 z}}{1 - \frac{1}{3 z}}$ by $\frac{3 z}{3 z}$ .

$\frac{3 z}{3 z} \cdot \frac{3 z - \frac{1}{3 z}}{1 - \frac{1}{3 z}}$

Combine.

$\frac{3 z (3 z - \frac{1}{3 z})}{3 z (1 - \frac{1}{3 z})}$

Step 2

Apply the distributive property.

$\frac{3 z (3 z) + 3 z (- \frac{1}{3 z})}{3 z \cdot 1 + 3 z (- \frac{1}{3 z})}$

Step 3

Simplify by cancelling.

Cancel the common factor of $3 z$ .

Move the leading negative in $- \frac{1}{3 z}$ into the numerator.

$\frac{3 z (3 z) + 3 z \frac{- 1}{3 z}}{3 z \cdot 1 + 3 z (- \frac{1}{3 z})}$

Cancel the common factor.

$\frac{3 z (3 z) + 3 z \frac{- 1}{3 z}}{3 z \cdot 1 + 3 z (- \frac{1}{3 z})}$

Rewrite the expression.

$\frac{3 z (3 z) - 1}{3 z \cdot 1 + 3 z (- \frac{1}{3 z})}$

Cancel the common factor of $3 z$ .

Move the leading negative in $- \frac{1}{3 z}$ into the numerator.

$\frac{3 z (3 z) - 1}{3 z \cdot 1 + 3 z \frac{- 1}{3 z}}$

Cancel the common factor.

$\frac{3 z (3 z) - 1}{3 z \cdot 1 + 3 z \frac{- 1}{3 z}}$

Rewrite the expression.

$\frac{3 z (3 z) - 1}{3 z \cdot 1 - 1}$

Step 4

Simplify the numerator.

Multiply $3$ by $3$ .

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

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

$\frac{9 (z^{1} z) - 1}{3 z \cdot 1 - 1}$

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

$\frac{9 (z^{1} z^{1}) - 1}{3 z \cdot 1 - 1}$

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

$\frac{9 z^{1 + 1} - 1}{3 z \cdot 1 - 1}$

Add $1$ and $1$ .

$\frac{9 z^{2} - 1}{3 z \cdot 1 - 1}$

Rewrite $9 z^{2}$ as ${(3 z)}^{2}$ .

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

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

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 z$ and $b = 1$ .

$\frac{(3 z + 1) (3 z - 1)}{3 z \cdot 1 - 1}$

Step 5

Reduce the expression by cancelling the common factors.

Multiply $3$ by $1$ .

$\frac{(3 z + 1) (3 z - 1)}{3 z - 1}$

Cancel the common factor of $3 z - 1$ .

Cancel the common factor.

$\frac{(3 z + 1) (3 z - 1)}{3 z - 1}$

Divide $3 z + 1$ by $1$ .

$3 z + 1$

Simplify (3z-1/(3z))/(1-1/(3z))

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