Precalculus Examples

$\frac{6 x {(x + 6)}^{4} - x^{2} \cdot 12 {(x + 6)}^{3}}{{(x + 6)}^{8}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $x {(x + 6)}^{3}$ out of $6 x {(x + 6)}^{4} - x^{2} \cdot (12 {(x + 6)}^{3})$ .

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Step 1.1.1

Factor $x {(x + 6)}^{3}$ out of $6 x {(x + 6)}^{4}$ .

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

Step 1.1.2

Factor $x {(x + 6)}^{3}$ out of $- x^{2} \cdot (12 {(x + 6)}^{3})$ .

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

Step 1.1.3

Factor $x {(x + 6)}^{3}$ out of $x {(x + 6)}^{3} (6 (x + 6)) + x {(x + 6)}^{3} (- x \cdot (12))$ .

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

Step 1.2

Multiply $12$ by $- 1$ .

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

Step 1.3

Simplify each term.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Multiply $6$ by $6$ .

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

Step 1.4

Subtract $12 x$ from $6 x$ .

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

Step 1.5

Factor $6$ out of $- 6 x + 36$ .

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Step 1.5.1

Factor $6$ out of $- 6 x$ .

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

Step 1.5.2

Factor $6$ out of $36$ .

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

Step 1.5.3

Factor $6$ out of $6 (- x) + 6 (6)$ .

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

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

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of ${(x + 6)}^{3}$ and ${(x + 6)}^{8}$ .

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Step 2.1.1

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

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

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor ${(x + 6)}^{3}$ out of ${(x + 6)}^{8}$ .

$\frac{{(x + 6)}^{3} (x \cdot 6 (- x + 6))}{{(x + 6)}^{3} {(x + 6)}^{5}}$

Step 2.1.2.2

Cancel the common factor.

$\frac{{(x + 6)}^{3} (x \cdot 6 (- x + 6))}{{(x + 6)}^{3} {(x + 6)}^{5}}$

Step 2.1.2.3

Rewrite the expression.

$\frac{x \cdot 6 (- x + 6)}{{(x + 6)}^{5}}$

Step 2.2

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

$\frac{6 \cdot x (- x + 6)}{{(x + 6)}^{5}}$

Step 2.3

Factor $- 1$ out of $- x$ .

$\frac{6 x (- (x) + 6)}{{(x + 6)}^{5}}$

Step 2.4

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

$\frac{6 x (- (x) - 1 (- 6))}{{(x + 6)}^{5}}$

Step 2.5

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

$\frac{6 x (- (x - 6))}{{(x + 6)}^{5}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

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

$\frac{6 x (- 1 (x - 6))}{{(x + 6)}^{5}}$

Step 2.6.2

Move the negative in front of the fraction.

$- \frac{(6 x) (x - 6)}{{(x + 6)}^{5}}$

Step 2.6.3

Reorder factors in $- \frac{(6 x) (x - 6)}{{(x + 6)}^{5}}$ .

$- \frac{6 x (x - 6)}{{(x + 6)}^{5}}$