Pre-Algebra Examples

$\frac{z^{2} - 9}{z} \div \frac{z + 3}{z - 3}$

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $9$ as $3^{2}$ .

$\frac{z^{2} - 3^{2}}{z} \cdot \frac{z - 3}{z + 3}$

Step 2.2

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 $b = 3$ .

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

Step 3

Cancel the common factor of $z + 3$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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

Step 4

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

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

Step 5

Simplify the numerator.

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Step 5.1

Raise $z - 3$ to the power of $1$ .

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

Step 5.2

Raise $z - 3$ to the power of $1$ .

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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

Step 6

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

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

Step 7

Expand $(z - 3) (z - 3)$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Apply the distributive property.

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

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $z$ by $z$ .

$\frac{z^{2} + z \cdot - 3 - 3 z - 3 \cdot - 3}{z}$

Step 8.1.2

Move $- 3$ to the left of $z$ .

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

Step 8.1.3

Multiply $- 3$ by $- 3$ .

$\frac{z^{2} - 3 z - 3 z + 9}{z}$

Step 8.2

Subtract $3 z$ from $- 3 z$ .

$\frac{z^{2} - 6 z + 9}{z}$

Step 9

Split the fraction $\frac{z^{2} - 6 z + 9}{z}$ into two fractions.

$\frac{z^{2} - 6 z}{z} + \frac{9}{z}$

Step 10

Split the fraction $\frac{z^{2} - 6 z}{z}$ into two fractions.

$\frac{z^{2}}{z} + \frac{- 6 z}{z} + \frac{9}{z}$

Step 11

Cancel the common factor of $z^{2}$ and $z$ .

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Step 11.1

Factor $z$ out of $z^{2}$ .

$\frac{z \cdot z}{z} + \frac{- 6 z}{z} + \frac{9}{z}$

Step 11.2

Cancel the common factors.

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Step 11.2.1

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

$\frac{z \cdot z}{z^{1}} + \frac{- 6 z}{z} + \frac{9}{z}$

Step 11.2.2

Factor $z$ out of $z^{1}$ .

$\frac{z \cdot z}{z \cdot 1} + \frac{- 6 z}{z} + \frac{9}{z}$

Step 11.2.3

Cancel the common factor.

$\frac{z \cdot z}{z \cdot 1} + \frac{- 6 z}{z} + \frac{9}{z}$

Step 11.2.4

Rewrite the expression.

$\frac{z}{1} + \frac{- 6 z}{z} + \frac{9}{z}$

Step 11.2.5

Divide $z$ by $1$ .

$z + \frac{- 6 z}{z} + \frac{9}{z}$

Step 12

Cancel the common factor of $z$ .

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Step 12.1

Cancel the common factor.

$z + \frac{- 6 z}{z} + \frac{9}{z}$

Step 12.2

Divide $- 6$ by $1$ .

$z - 6 + \frac{9}{z}$