Algebra Examples

$\frac{\frac{x - 6}{6} - \frac{x - 2}{x - 6}}{\frac{36}{x - 2} + \frac{4}{9}}$

Step 1

Multiply the numerator and denominator of the fraction by $18 (x - 6) (x - 2)$ .

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Step 1.1

Multiply $\frac{\frac{x - 6}{6} - \frac{x - 2}{x - 6}}{\frac{36}{x - 2} + \frac{4}{9}}$ by $\frac{18 (x - 6) (x - 2)}{18 (x - 6) (x - 2)}$ .

$\frac{18 (x - 6) (x - 2)}{18 (x - 6) (x - 2)} \cdot \frac{\frac{x - 6}{6} - \frac{x - 2}{x - 6}}{\frac{36}{x - 2} + \frac{4}{9}}$

Step 1.2

Combine.

$\frac{18 (x - 6) (x - 2) (\frac{x - 6}{6} - \frac{x - 2}{x - 6})}{18 (x - 6) (x - 2) (\frac{36}{x - 2} + \frac{4}{9})}$

Step 2

Apply the distributive property.

$\frac{18 (x - 6) (x - 2) \frac{x - 6}{6} + 18 (x - 6) (x - 2) (- \frac{x - 2}{x - 6})}{18 (x - 6) (x - 2) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $6$ .

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Step 3.1.1

Factor $6$ out of $18 (x - 6) (x - 2)$ .

$\frac{6 (3 (x - 6) (x - 2)) \frac{x - 6}{6} + 18 (x - 6) (x - 2) (- \frac{x - 2}{x - 6})}{18 (x - 6) (x - 2) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.1.2

Cancel the common factor.

$\frac{6 (3 (x - 6) (x - 2)) \frac{x - 6}{6} + 18 (x - 6) (x - 2) (- \frac{x - 2}{x - 6})}{18 (x - 6) (x - 2) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.1.3

Rewrite the expression.

$\frac{3 (x - 6) (x - 2) (x - 6) + 18 (x - 6) (x - 2) (- \frac{x - 2}{x - 6})}{18 (x - 6) (x - 2) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.2

Raise $x - 6$ to the power of $1$ .

$\frac{3 ({(x - 6)}^{1} (x - 6)) (x - 2) + 18 (x - 6) (x - 2) (- \frac{x - 2}{x - 6})}{18 (x - 6) (x - 2) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.3

Raise $x - 6$ to the power of $1$ .

$\frac{3 ({(x - 6)}^{1} {(x - 6)}^{1}) (x - 2) + 18 (x - 6) (x - 2) (- \frac{x - 2}{x - 6})}{18 (x - 6) (x - 2) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.4

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

$\frac{3 {(x - 6)}^{1 + 1} (x - 2) + 18 (x - 6) (x - 2) (- \frac{x - 2}{x - 6})}{18 (x - 6) (x - 2) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.5

Add $1$ and $1$ .

$\frac{3 {(x - 6)}^{2} (x - 2) + 18 (x - 6) (x - 2) (- \frac{x - 2}{x - 6})}{18 (x - 6) (x - 2) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.6

Cancel the common factor of $x - 6$ .

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Step 3.6.1

Move the leading negative in $- \frac{x - 2}{x - 6}$ into the numerator.

$\frac{3 {(x - 6)}^{2} (x - 2) + 18 (x - 6) (x - 2) \frac{- (x - 2)}{x - 6}}{18 (x - 6) (x - 2) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.6.2

Factor $x - 6$ out of $18 (x - 6) (x - 2)$ .

$\frac{3 {(x - 6)}^{2} (x - 2) + (x - 6) (18 (x - 2)) \frac{- (x - 2)}{x - 6}}{18 (x - 6) (x - 2) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.6.3

Cancel the common factor.

$\frac{3 {(x - 6)}^{2} (x - 2) + (x - 6) (18 (x - 2)) \frac{- (x - 2)}{x - 6}}{18 (x - 6) (x - 2) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.6.4

Rewrite the expression.

$\frac{3 {(x - 6)}^{2} (x - 2) + 18 (x - 2) (- (x - 2))}{18 (x - 6) (x - 2) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.7

Multiply $- 1$ by $18$ .

$\frac{3 {(x - 6)}^{2} (x - 2) - 18 (x - 2) (x - 2)}{18 (x - 6) (x - 2) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.8

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

$\frac{3 {(x - 6)}^{2} (x - 2) - 18 ({(x - 2)}^{1} (x - 2))}{18 (x - 6) (x - 2) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.9

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

$\frac{3 {(x - 6)}^{2} (x - 2) - 18 ({(x - 2)}^{1} {(x - 2)}^{1})}{18 (x - 6) (x - 2) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.10

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

$\frac{3 {(x - 6)}^{2} (x - 2) - 18 {(x - 2)}^{1 + 1}}{18 (x - 6) (x - 2) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.11

Add $1$ and $1$ .

$\frac{3 {(x - 6)}^{2} (x - 2) - 18 {(x - 2)}^{2}}{18 (x - 6) (x - 2) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.12

Cancel the common factor of $x - 2$ .

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Step 3.12.1

Factor $x - 2$ out of $18 (x - 6) (x - 2)$ .

$\frac{3 {(x - 6)}^{2} (x - 2) - 18 {(x - 2)}^{2}}{(x - 2) (18 (x - 6)) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.12.2

Cancel the common factor.

$\frac{3 {(x - 6)}^{2} (x - 2) - 18 {(x - 2)}^{2}}{(x - 2) (18 (x - 6)) \frac{36}{x - 2} + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.12.3

Rewrite the expression.

$\frac{3 {(x - 6)}^{2} (x - 2) - 18 {(x - 2)}^{2}}{18 (x - 6) \cdot 36 + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.13

Multiply $36$ by $18$ .

$\frac{3 {(x - 6)}^{2} (x - 2) - 18 {(x - 2)}^{2}}{648 (x - 6) + 18 (x - 6) (x - 2) \frac{4}{9}}$

Step 3.14

Cancel the common factor of $9$ .

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Step 3.14.1

Factor $9$ out of $18 (x - 6) (x - 2)$ .

$\frac{3 {(x - 6)}^{2} (x - 2) - 18 {(x - 2)}^{2}}{648 (x - 6) + 9 (2 (x - 6) (x - 2)) \frac{4}{9}}$

Step 3.14.2

Cancel the common factor.

$\frac{3 {(x - 6)}^{2} (x - 2) - 18 {(x - 2)}^{2}}{648 (x - 6) + 9 (2 (x - 6) (x - 2)) \frac{4}{9}}$

Step 3.14.3

Rewrite the expression.

$\frac{3 {(x - 6)}^{2} (x - 2) - 18 {(x - 2)}^{2}}{648 (x - 6) + 2 (x - 6) (x - 2) \cdot 4}$

Step 3.15

Multiply $4$ by $2$ .

$\frac{3 {(x - 6)}^{2} (x - 2) - 18 {(x - 2)}^{2}}{648 (x - 6) + 8 (x - 6) (x - 2)}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $3 (x - 2)$ out of $3 {(x - 6)}^{2} (x - 2) - 18 {(x - 2)}^{2}$ .

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Step 4.1.1

Factor $3 (x - 2)$ out of $3 {(x - 6)}^{2} (x - 2)$ .

$\frac{3 (x - 2) ({(x - 6)}^{2}) - 18 {(x - 2)}^{2}}{648 (x - 6) + 8 (x - 6) (x - 2)}$

Step 4.1.2

Factor $3 (x - 2)$ out of $- 18 {(x - 2)}^{2}$ .

$\frac{3 (x - 2) ({(x - 6)}^{2}) + 3 (x - 2) (- 6 (x - 2))}{648 (x - 6) + 8 (x - 6) (x - 2)}$

Step 4.1.3

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

$\frac{3 (x - 2) ({(x - 6)}^{2} - 6 (x - 2))}{648 (x - 6) + 8 (x - 6) (x - 2)}$

Step 4.2

Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

$\frac{3 (x - 2) ((x - 6) (x - 6) - 6 (x - 2))}{648 (x - 6) + 8 (x - 6) (x - 2)}$

Step 4.3

Expand $(x - 6) (x - 6)$ using the FOIL Method.

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Step 4.3.1

Apply the distributive property.

$\frac{3 (x - 2) (x (x - 6) - 6 (x - 6) - 6 (x - 2))}{648 (x - 6) + 8 (x - 6) (x - 2)}$

Step 4.3.2

Apply the distributive property.

$\frac{3 (x - 2) (x \cdot x + x \cdot - 6 - 6 (x - 6) - 6 (x - 2))}{648 (x - 6) + 8 (x - 6) (x - 2)}$

Step 4.3.3

Apply the distributive property.

$\frac{3 (x - 2) (x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6 - 6 (x - 2))}{648 (x - 6) + 8 (x - 6) (x - 2)}$

Step 4.4

Simplify and combine like terms.

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Step 4.4.1

Simplify each term.

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Step 4.4.1.1

Multiply $x$ by $x$ .

$\frac{3 (x - 2) (x^{2} + x \cdot - 6 - 6 x - 6 \cdot - 6 - 6 (x - 2))}{648 (x - 6) + 8 (x - 6) (x - 2)}$

Step 4.4.1.2

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

$\frac{3 (x - 2) (x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6 - 6 (x - 2))}{648 (x - 6) + 8 (x - 6) (x - 2)}$

Step 4.4.1.3

Multiply $- 6$ by $- 6$ .

$\frac{3 (x - 2) (x^{2} - 6 x - 6 x + 36 - 6 (x - 2))}{648 (x - 6) + 8 (x - 6) (x - 2)}$

Step 4.4.2

Subtract $6 x$ from $- 6 x$ .

$\frac{3 (x - 2) (x^{2} - 12 x + 36 - 6 (x - 2))}{648 (x - 6) + 8 (x - 6) (x - 2)}$

Step 4.5

Apply the distributive property.

$\frac{3 (x - 2) (x^{2} - 12 x + 36 - 6 x - 6 \cdot - 2)}{648 (x - 6) + 8 (x - 6) (x - 2)}$

Step 4.6

Multiply $- 6$ by $- 2$ .

$\frac{3 (x - 2) (x^{2} - 12 x + 36 - 6 x + 12)}{648 (x - 6) + 8 (x - 6) (x - 2)}$

Step 4.7

Subtract $6 x$ from $- 12 x$ .

$\frac{3 (x - 2) (x^{2} - 18 x + 36 + 12)}{648 (x - 6) + 8 (x - 6) (x - 2)}$

Step 4.8

Add $36$ and $12$ .

$\frac{3 (x - 2) (x^{2} - 18 x + 48)}{648 (x - 6) + 8 (x - 6) (x - 2)}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $8 (x - 6)$ out of $648 (x - 6) + 8 (x - 6) (x - 2)$ .

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Step 5.1.1

Factor $8 (x - 6)$ out of $648 (x - 6)$ .

$\frac{3 (x - 2) (x^{2} - 18 x + 48)}{8 (x - 6) (81) + 8 (x - 6) (x - 2)}$

Step 5.1.2

Factor $8 (x - 6)$ out of $8 (x - 6) (81) + 8 (x - 6) (x - 2)$ .

$\frac{3 (x - 2) (x^{2} - 18 x + 48)}{8 (x - 6) (81 + x - 2)}$

Step 5.2

Subtract $2$ from $81$ .

$\frac{3 (x - 2) (x^{2} - 18 x + 48)}{8 (x - 6) (x + 79)}$