Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Factor $x {(x + 6)}^{3}$ out of $4 x {(x + 6)}^{4} - x^{2} \cdot (8 {(x + 6)}^{3})$ .

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Step 1.1.1

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

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

Step 1.1.2

Factor $x {(x + 6)}^{3}$ out of $- x^{2} \cdot (8 {(x + 6)}^{3})$ .

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

Step 1.1.3

Factor $x {(x + 6)}^{3}$ out of $x {(x + 6)}^{3} (4 (x + 6)) + x {(x + 6)}^{3} (- x \cdot (8))$ .

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

Step 1.2

Multiply $8$ by $- 1$ .

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

Step 1.3

Simplify each term.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Multiply $4$ by $6$ .

$\frac{x {(x + 6)}^{3} (4 x + 24 - 8 x)}{{(x + 6)}^{8}}$

Step 1.4

Subtract $8 x$ from $4 x$ .

$\frac{x {(x + 6)}^{3} (- 4 x + 24)}{{(x + 6)}^{8}}$

Step 1.5

Factor $4$ out of $- 4 x + 24$ .

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Step 1.5.1

Factor $4$ out of $- 4 x$ .

$\frac{x {(x + 6)}^{3} (4 (- x) + 24)}{{(x + 6)}^{8}}$

Step 1.5.2

Factor $4$ out of $24$ .

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

Step 1.5.3

Factor $4$ out of $4 (- x) + 4 (6)$ .

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

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

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 $x {(x + 6)}^{3} \cdot 4 (- x + 6)$ .

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

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

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

$\frac{{(x + 6)}^{3} (x \cdot 4 (- x + 6))}{{(x + 6)}^{3} {(x + 6)}^{5}}$

Step 2.1.2.2

Cancel the common factor.

$\frac{{(x + 6)}^{3} (x \cdot 4 (- x + 6))}{{(x + 6)}^{3} {(x + 6)}^{5}}$

Step 2.1.2.3

Rewrite the expression.

$\frac{x \cdot 4 (- x + 6)}{{(x + 6)}^{5}}$

Step 2.2

Move $4$ to the left of $x$ .

$\frac{4 \cdot x (- x + 6)}{{(x + 6)}^{5}}$

Step 2.3

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

$\frac{4 x (- (x) + 6)}{{(x + 6)}^{5}}$

Step 2.4

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

$\frac{4 x (- (x) - 1 (- 6))}{{(x + 6)}^{5}}$

Step 2.5

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

$\frac{4 x (- (x - 6))}{{(x + 6)}^{5}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

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

$\frac{4 x (- 1 (x - 6))}{{(x + 6)}^{5}}$

Step 2.6.2

Move the negative in front of the fraction.

$- \frac{(4 x) (x - 6)}{{(x + 6)}^{5}}$

Step 2.6.3

Reorder factors in $- \frac{(4 x) (x - 6)}{{(x + 6)}^{5}}$ .

$- \frac{4 x (x - 6)}{{(x + 6)}^{5}}$