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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