Pre-Algebra Examples

$\frac{f^{2} - 9}{- f^{3} + 6 f^{2}} \div \frac{- 9 f - 27}{f^{2} - 12 f + 36}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{f^{2} - 9}{- f^{3} + 6 f^{2}} \cdot \frac{f^{2} - 12 f + 36}{- 9 f - 27}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{f^{2} - 3^{2}}{- f^{3} + 6 f^{2}} \cdot \frac{f^{2} - 12 f + 36}{- 9 f - 27}$

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 $a = f$ and $b = 3$ .

$\frac{(f + 3) (f - 3)}{- f^{3} + 6 f^{2}} \cdot \frac{f^{2} - 12 f + 36}{- 9 f - 27}$

Step 3

Factor $f^{2}$ out of $- f^{3} + 6 f^{2}$ .

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Step 3.1

Factor $f^{2}$ out of $- f^{3}$ .

$\frac{(f + 3) (f - 3)}{f^{2} (- f) + 6 f^{2}} \cdot \frac{f^{2} - 12 f + 36}{- 9 f - 27}$

Step 3.2

Factor $f^{2}$ out of $6 f^{2}$ .

$\frac{(f + 3) (f - 3)}{f^{2} (- f) + f^{2} \cdot 6} \cdot \frac{f^{2} - 12 f + 36}{- 9 f - 27}$

Step 3.3

Factor $f^{2}$ out of $f^{2} (- f) + f^{2} \cdot 6$ .

$\frac{(f + 3) (f - 3)}{f^{2} (- f + 6)} \cdot \frac{f^{2} - 12 f + 36}{- 9 f - 27}$

Step 4

Factor using the perfect square rule.

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Step 4.1

Rewrite $36$ as $6^{2}$ .

$\frac{(f + 3) (f - 3)}{f^{2} (- f + 6)} \cdot \frac{f^{2} - 12 f + 6^{2}}{- 9 f - 27}$

Step 4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$12 f = 2 \cdot f \cdot 6$

Step 4.3

Rewrite the polynomial.

$\frac{(f + 3) (f - 3)}{f^{2} (- f + 6)} \cdot \frac{f^{2} - 2 \cdot f \cdot 6 + 6^{2}}{- 9 f - 27}$

Step 4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = f$ and $b = 6$ .

$\frac{(f + 3) (f - 3)}{f^{2} (- f + 6)} \cdot \frac{{(f - 6)}^{2}}{- 9 f - 27}$

Step 5

Factor $9$ out of $- 9 f - 27$ .

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Step 5.1

Factor $9$ out of $- 9 f$ .

$\frac{(f + 3) (f - 3)}{f^{2} (- f + 6)} \cdot \frac{{(f - 6)}^{2}}{9 (- f) - 27}$

Step 5.2

Factor $9$ out of $- 27$ .

$\frac{(f + 3) (f - 3)}{f^{2} (- f + 6)} \cdot \frac{{(f - 6)}^{2}}{9 (- f) + 9 (- 3)}$

Step 5.3

Factor $9$ out of $9 (- f) + 9 (- 3)$ .

$\frac{(f + 3) (f - 3)}{f^{2} (- f + 6)} \cdot \frac{{(f - 6)}^{2}}{9 (- f - 3)}$

Step 6

Combine.

$\frac{(f + 3) (f - 3) {(f - 6)}^{2}}{f^{2} (- f + 6) (9 (- f - 3))}$

Step 7

Cancel the common factor of $f + 3$ and $- f - 3$ .

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Step 7.1

Factor $- 1$ out of $f$ .

$\frac{(- 1 (- f) + 3) (f - 3) {(f - 6)}^{2}}{f^{2} (- f + 6) (9 (- f - 3))}$

Step 7.2

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{(- 1 (- f) - 1 (- 3)) (f - 3) {(f - 6)}^{2}}{f^{2} (- f + 6) (9 (- f - 3))}$

Step 7.3

Factor $- 1$ out of $- 1 (- f) - 1 (- 3)$ .

$\frac{- 1 (- f - 3) (f - 3) {(f - 6)}^{2}}{f^{2} (- f + 6) (9 (- f - 3))}$

Step 7.4

Cancel the common factor.

$\frac{- 1 (- f - 3) (f - 3) {(f - 6)}^{2}}{f^{2} (- f + 6) (9 (- f - 3))}$

Step 7.5

Rewrite the expression.

$\frac{(- 1 (f - 3)) {(f - 6)}^{2}}{f^{2} (- f + 6) \cdot (9)}$

Step 8

Cancel the common factor of ${(f - 6)}^{2}$ and $- f + 6$ .

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Step 8.1

Factor $- 1$ out of $f$ .

$\frac{- 1 (f - 3) {(- 1 (- f) - 6)}^{2}}{f^{2} (- f + 6) \cdot (9)}$

Step 8.2

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{- 1 (f - 3) {(- 1 (- f) - 1 (6))}^{2}}{f^{2} (- f + 6) \cdot (9)}$

Step 8.3

Factor $- 1$ out of $- 1 (- f) - 1 (6)$ .

$\frac{- 1 (f - 3) {(- 1 (- f + 6))}^{2}}{f^{2} (- f + 6) \cdot (9)}$

Step 8.4

Apply the product rule to $- 1 (- f + 6)$ .

$\frac{- 1 (f - 3) ({(- 1)}^{2} {(- f + 6)}^{2})}{f^{2} (- f + 6) \cdot (9)}$

Step 8.5

Raise $- 1$ to the power of $2$ .

$\frac{- 1 (f - 3) (1 {(- f + 6)}^{2})}{f^{2} (- f + 6) \cdot (9)}$

Step 8.6

Multiply ${(- f + 6)}^{2}$ by $1$ .

$\frac{- 1 (f - 3) {(- f + 6)}^{2}}{f^{2} (- f + 6) \cdot (9)}$

Step 8.7

Factor $- f + 6$ out of $- 1 (f - 3) {(- f + 6)}^{2}$ .

$\frac{(- f + 6) (- 1 (f - 3) (- f + 6))}{f^{2} (- f + 6) \cdot 9}$

Step 8.8

Cancel the common factors.

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Step 8.8.1

Factor $- f + 6$ out of $f^{2} (- f + 6) \cdot 9$ .

$\frac{(- f + 6) (- 1 (f - 3) (- f + 6))}{(- f + 6) (f^{2} \cdot 9)}$

Step 8.8.2

Cancel the common factor.

$\frac{(- f + 6) (- 1 (f - 3) (- f + 6))}{(- f + 6) (f^{2} \cdot 9)}$

Step 8.8.3

Rewrite the expression.

$\frac{- 1 (f - 3) (- f + 6)}{f^{2} \cdot 9}$

Step 9

Move $9$ to the left of $f^{2}$ .

$\frac{- 1 (f - 3) (- f + 6)}{9 \cdot f^{2}}$

Step 10

Move the negative in front of the fraction.

$- \frac{(f - 3) (- f + 6)}{9 \cdot f^{2}}$

Step 11

Factor $- 1$ out of $- f$ .

$- \frac{(f - 3) (- (f) + 6)}{9 f^{2}}$

Step 12

Rewrite $6$ as $- 1 (- 6)$ .

$- \frac{(f - 3) (- (f) - 1 (- 6))}{9 f^{2}}$

Step 13

Factor $- 1$ out of $- (f) - 1 (- 6)$ .

$- \frac{(f - 3) (- (f - 6))}{9 f^{2}}$

Step 14

Rewrite $- (f - 6)$ as $- 1 (f - 6)$ .

$- \frac{(f - 3) (- 1 (f - 6))}{9 f^{2}}$

Step 15

Move the negative in front of the fraction.

$- - \frac{(f - 3) (f - 6)}{9 f^{2}}$

Step 16

Multiply $- 1$ by $- 1$ .

$1 \frac{(f - 3) (f - 6)}{9 f^{2}}$

Step 17

Multiply $\frac{(f - 3) (f - 6)}{9 f^{2}}$ by $1$ .

$\frac{(f - 3) (f - 6)}{9 f^{2}}$

Step 18

Expand $(f - 3) (f - 6)$ using the FOIL Method.

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Step 18.1

Apply the distributive property.

$\frac{f (f - 6) - 3 (f - 6)}{9 f^{2}}$

Step 18.2

Apply the distributive property.

$\frac{f \cdot f + f \cdot - 6 - 3 (f - 6)}{9 f^{2}}$

Step 18.3

Apply the distributive property.

$\frac{f \cdot f + f \cdot - 6 - 3 f - 3 \cdot - 6}{9 f^{2}}$

Step 19

Simplify and combine like terms.

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Step 19.1

Simplify each term.

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Step 19.1.1

Multiply $f$ by $f$ .

$\frac{f^{2} + f \cdot - 6 - 3 f - 3 \cdot - 6}{9 f^{2}}$

Step 19.1.2

Move $- 6$ to the left of $f$ .

$\frac{f^{2} - 6 \cdot f - 3 f - 3 \cdot - 6}{9 f^{2}}$

Step 19.1.3

Multiply $- 3$ by $- 6$ .

$\frac{f^{2} - 6 f - 3 f + 18}{9 f^{2}}$

Step 19.2

Subtract $3 f$ from $- 6 f$ .

$\frac{f^{2} - 9 f + 18}{9 f^{2}}$

Step 20

Split the fraction $\frac{f^{2} - 9 f + 18}{9 f^{2}}$ into two fractions.

$\frac{f^{2} - 9 f}{9 f^{2}} + \frac{18}{9 f^{2}}$

Step 21

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

$\frac{f^{2}}{9 f^{2}} + \frac{- 9 f}{9 f^{2}} + \frac{18}{9 f^{2}}$

Step 22

Cancel the common factor of $f^{2}$ .

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Step 22.1

Cancel the common factor.

$\frac{f^{2}}{9 f^{2}} + \frac{- 9 f}{9 f^{2}} + \frac{18}{9 f^{2}}$

Step 22.2

Rewrite the expression.

$\frac{1}{9} + \frac{- 9 f}{9 f^{2}} + \frac{18}{9 f^{2}}$

Step 23

Cancel the common factor of $- 9$ and $9$ .

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Step 23.1

Factor $9$ out of $- 9 f$ .

$\frac{1}{9} + \frac{9 (- f)}{9 f^{2}} + \frac{18}{9 f^{2}}$

Step 23.2

Cancel the common factors.

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Step 23.2.1

Factor $9$ out of $9 f^{2}$ .

$\frac{1}{9} + \frac{9 (- f)}{9 (f^{2})} + \frac{18}{9 f^{2}}$

Step 23.2.2

Cancel the common factor.

$\frac{1}{9} + \frac{9 (- f)}{9 f^{2}} + \frac{18}{9 f^{2}}$

Step 23.2.3

Rewrite the expression.

$\frac{1}{9} + \frac{- f}{f^{2}} + \frac{18}{9 f^{2}}$

Step 24

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

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Step 24.1

Factor $f$ out of $- f$ .

$\frac{1}{9} + \frac{f \cdot - 1}{f^{2}} + \frac{18}{9 f^{2}}$

Step 24.2

Cancel the common factors.

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Step 24.2.1

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

$\frac{1}{9} + \frac{f \cdot - 1}{f \cdot f} + \frac{18}{9 f^{2}}$

Step 24.2.2

Cancel the common factor.

$\frac{1}{9} + \frac{f \cdot - 1}{f \cdot f} + \frac{18}{9 f^{2}}$

Step 24.2.3

Rewrite the expression.

$\frac{1}{9} + \frac{- 1}{f} + \frac{18}{9 f^{2}}$

Step 25

Move the negative in front of the fraction.

$\frac{1}{9} - \frac{1}{f} + \frac{18}{9 f^{2}}$

Step 26

Cancel the common factor of $18$ and $9$ .

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Step 26.1

Factor $9$ out of $18$ .

$\frac{1}{9} - \frac{1}{f} + \frac{9 \cdot 2}{9 f^{2}}$

Step 26.2

Cancel the common factors.

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Step 26.2.1

Factor $9$ out of $9 f^{2}$ .

$\frac{1}{9} - \frac{1}{f} + \frac{9 \cdot 2}{9 (f^{2})}$

Step 26.2.2

Cancel the common factor.

$\frac{1}{9} - \frac{1}{f} + \frac{9 \cdot 2}{9 f^{2}}$

Step 26.2.3

Rewrite the expression.

$\frac{1}{9} - \frac{1}{f} + \frac{2}{f^{2}}$