Calculus Examples

$- \frac{2 k}{x^{3}} + \frac{16 k}{{(12 - x)}^{3}}$

Step 1

To write $- \frac{2 k}{x^{3}}$ as a fraction with a common denominator, multiply by $\frac{{(12 - x)}^{3}}{{(12 - x)}^{3}}$ .

$- \frac{2 k}{x^{3}} \cdot \frac{{(12 - x)}^{3}}{{(12 - x)}^{3}} + \frac{16 k}{{(12 - x)}^{3}}$

Step 2

To write $\frac{16 k}{{(12 - x)}^{3}}$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

$- \frac{2 k}{x^{3}} \cdot \frac{{(12 - x)}^{3}}{{(12 - x)}^{3}} + \frac{16 k}{{(12 - x)}^{3}} \cdot \frac{x^{3}}{x^{3}}$

Step 3

Write each expression with a common denominator of $x^{3} {(12 - x)}^{3}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{2 k}{x^{3}}$ by $\frac{{(12 - x)}^{3}}{{(12 - x)}^{3}}$ .

$- \frac{2 k {(12 - x)}^{3}}{x^{3} {(12 - x)}^{3}} + \frac{16 k}{{(12 - x)}^{3}} \cdot \frac{x^{3}}{x^{3}}$

Step 3.2

Multiply $\frac{16 k}{{(12 - x)}^{3}}$ by $\frac{x^{3}}{x^{3}}$ .

$- \frac{2 k {(12 - x)}^{3}}{x^{3} {(12 - x)}^{3}} + \frac{16 k x^{3}}{{(12 - x)}^{3} x^{3}}$

Step 3.3

Reorder the factors of ${(12 - x)}^{3} x^{3}$ .

$- \frac{2 k {(12 - x)}^{3}}{x^{3} {(12 - x)}^{3}} + \frac{16 k x^{3}}{x^{3} {(12 - x)}^{3}}$

Step 4

Combine the numerators over the common denominator.

$\frac{- 2 k {(12 - x)}^{3} + 16 k x^{3}}{x^{3} {(12 - x)}^{3}}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $2 k$ out of $- 2 k {(12 - x)}^{3} + 16 k x^{3}$ .

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Step 5.1.1

Factor $2 k$ out of $- 2 k {(12 - x)}^{3}$ .

$\frac{2 k (- {(12 - x)}^{3}) + 16 k x^{3}}{x^{3} {(12 - x)}^{3}}$

Step 5.1.2

Factor $2 k$ out of $16 k x^{3}$ .

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

Step 5.1.3

Factor $2 k$ out of $2 k (- {(12 - x)}^{3}) + 2 k (8 x^{3})$ .

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

Step 5.2

Rewrite $- {(12 - x)}^{3}$ as ${(- (12 - x))}^{3}$ .

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

Step 5.3

Rewrite $8 x^{3}$ as ${(2 x)}^{3}$ .

$\frac{2 k ({(- (12 - x))}^{3} + {(2 x)}^{3})}{x^{3} {(12 - x)}^{3}}$

Step 5.4

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = - (12 - x)$ and $b = 2 x$ .

$\frac{2 k ((- (12 - x) + 2 x) ({(- (12 - x))}^{2} - - (12 - x) (2 x) + {(2 x)}^{2}))}{x^{3} {(12 - x)}^{3}}$

Step 5.5

Simplify.

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Step 5.5.1

Apply the distributive property.

$\frac{2 k ((- 1 \cdot 12 - - x + 2 x) ({(- (12 - x))}^{2} - - (12 - x) (2 x) + {(2 x)}^{2}))}{x^{3} {(12 - x)}^{3}}$

Step 5.5.2

Multiply $- 1$ by $12$ .

$\frac{2 k ((- 12 - - x + 2 x) ({(- (12 - x))}^{2} - - (12 - x) (2 x) + {(2 x)}^{2}))}{x^{3} {(12 - x)}^{3}}$

Step 5.5.3

Multiply $- - x$ .

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Step 5.5.3.1

Multiply $- 1$ by $- 1$ .

$\frac{2 k ((- 12 + 1 x + 2 x) ({(- (12 - x))}^{2} - - (12 - x) (2 x) + {(2 x)}^{2}))}{x^{3} {(12 - x)}^{3}}$

Step 5.5.3.2

Multiply $x$ by $1$ .

$\frac{2 k ((- 12 + x + 2 x) ({(- (12 - x))}^{2} - - (12 - x) (2 x) + {(2 x)}^{2}))}{x^{3} {(12 - x)}^{3}}$

Step 5.5.4

Add $x$ and $2 x$ .

$\frac{2 k ((- 12 + 3 x) ({(- (12 - x))}^{2} - - (12 - x) (2 x) + {(2 x)}^{2}))}{x^{3} {(12 - x)}^{3}}$

Step 5.5.5

Factor $3$ out of $- 12 + 3 x$ .

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Step 5.5.5.1

Factor $3$ out of $- 12$ .

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

Step 5.5.5.2

Factor $3$ out of $3 \cdot - 4 + 3 x$ .

$\frac{2 k (3 (- 4 + x) ({(- (12 - x))}^{2} - - (12 - x) (2 x) + {(2 x)}^{2}))}{x^{3} {(12 - x)}^{3}}$

Step 5.5.6

Combine exponents.

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Step 5.5.6.1

Multiply $- 1$ by $- 1$ .

$\frac{2 k (3 (- 4 + x) ({(- (12 - x))}^{2} + 1 (12 - x) (2 x) + {(2 x)}^{2}))}{x^{3} {(12 - x)}^{3}}$

Step 5.5.6.2

Multiply $12 - x$ by $1$ .

$\frac{2 k (3 (- 4 + x) ({(- (12 - x))}^{2} + (12 - x) (2 x) + {(2 x)}^{2}))}{x^{3} {(12 - x)}^{3}}$

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

Step 5.6

Multiply $3$ by $2$ .

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

Step 5.7

Simplify each term.

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Step 5.7.1

Apply the distributive property.

$\frac{6 k (- 4 + x) ({(- 1 \cdot 12 - - x)}^{2} + (12 - x) (2 x) + {(2 x)}^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.7.2

Multiply $- 1$ by $12$ .

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

Step 5.7.3

Multiply $- - x$ .

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Step 5.7.3.1

Multiply $- 1$ by $- 1$ .

$\frac{6 k (- 4 + x) ({(- 12 + 1 x)}^{2} + (12 - x) (2 x) + {(2 x)}^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.7.3.2

Multiply $x$ by $1$ .

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

Step 5.7.4

Rewrite ${(- 12 + x)}^{2}$ as $(- 12 + x) (- 12 + x)$ .

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

Step 5.7.5

Expand $(- 12 + x) (- 12 + x)$ using the FOIL Method.

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Step 5.7.5.1

Apply the distributive property.

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

Step 5.7.5.2

Apply the distributive property.

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

Step 5.7.5.3

Apply the distributive property.

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

Step 5.7.6

Simplify and combine like terms.

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Step 5.7.6.1

Simplify each term.

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Step 5.7.6.1.1

Multiply $- 12$ by $- 12$ .

$\frac{6 k (- 4 + x) (144 - 12 x + x \cdot - 12 + x \cdot x + (12 - x) (2 x) + {(2 x)}^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.7.6.1.2

Move $- 12$ to the left of $x$ .

$\frac{6 k (- 4 + x) (144 - 12 x - 12 \cdot x + x \cdot x + (12 - x) (2 x) + {(2 x)}^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.7.6.1.3

Multiply $x$ by $x$ .

$\frac{6 k (- 4 + x) (144 - 12 x - 12 x + x^{2} + (12 - x) (2 x) + {(2 x)}^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.7.6.2

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

$\frac{6 k (- 4 + x) (144 - 24 x + x^{2} + (12 - x) (2 x) + {(2 x)}^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.7.7

Apply the distributive property.

$\frac{6 k (- 4 + x) (144 - 24 x + x^{2} + 12 (2 x) - x (2 x) + {(2 x)}^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.7.8

Multiply $2$ by $12$ .

$\frac{6 k (- 4 + x) (144 - 24 x + x^{2} + 24 x - x (2 x) + {(2 x)}^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.7.9

Rewrite using the commutative property of multiplication.

$\frac{6 k (- 4 + x) (144 - 24 x + x^{2} + 24 x - 1 \cdot 2 x \cdot x + {(2 x)}^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.7.10

Simplify each term.

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Step 5.7.10.1

Multiply $x$ by $x$ by adding the exponents.

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Step 5.7.10.1.1

Move $x$ .

$\frac{6 k (- 4 + x) (144 - 24 x + x^{2} + 24 x - 1 \cdot 2 (x \cdot x) + {(2 x)}^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.7.10.1.2

Multiply $x$ by $x$ .

$\frac{6 k (- 4 + x) (144 - 24 x + x^{2} + 24 x - 1 \cdot 2 x^{2} + {(2 x)}^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.7.10.2

Multiply $- 1$ by $2$ .

$\frac{6 k (- 4 + x) (144 - 24 x + x^{2} + 24 x - 2 x^{2} + {(2 x)}^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.7.11

Apply the product rule to $2 x$ .

$\frac{6 k (- 4 + x) (144 - 24 x + x^{2} + 24 x - 2 x^{2} + 2^{2} x^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.7.12

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

$\frac{6 k (- 4 + x) (144 - 24 x + x^{2} + 24 x - 2 x^{2} + 4 x^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.8

Add $- 24 x$ and $24 x$ .

$\frac{6 k (- 4 + x) (144 + x^{2} + 0 x - 2 x^{2} + 4 x^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.9

Multiply $0$ by $x$ .

$\frac{6 k (- 4 + x) (144 + x^{2} + 0 - 2 x^{2} + 4 x^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.10

Add $144 + x^{2}$ and $0$ .

$\frac{6 k (- 4 + x) (144 + x^{2} - 2 x^{2} + 4 x^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.11

Subtract $2 x^{2}$ from $x^{2}$ .

$\frac{6 k (- 4 + x) (144 - x^{2} + 4 x^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.12

Add $- x^{2}$ and $4 x^{2}$ .

$\frac{6 k (- 4 + x) (144 + 3 x^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.13

Factor $3$ out of $144 + 3 x^{2}$ .

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Step 5.13.1

Factor $3$ out of $144$ .

$\frac{6 k (- 4 + x) (3 \cdot 48 + 3 x^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.13.2

Factor $3$ out of $3 \cdot 48 + 3 x^{2}$ .

$\frac{6 k (- 4 + x) (3 (48 + x^{2}))}{x^{3} {(12 - x)}^{3}}$

$\frac{6 k (- 4 + x) \cdot 3 (48 + x^{2})}{x^{3} {(12 - x)}^{3}}$

Step 5.14

Multiply $3$ by $6$ .

$\frac{18 k (- 4 + x) (48 + x^{2})}{x^{3} {(12 - x)}^{3}}$

Step 6

Simplify with factoring out.

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Step 6.1

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{18 k (- 1 (4) + x) (48 + x^{2})}{x^{3} {(12 - x)}^{3}}$

Step 6.2

Factor $- 1$ out of $x$ .

$\frac{18 k (- 1 (4) - 1 (- x)) (48 + x^{2})}{x^{3} {(12 - x)}^{3}}$

Step 6.3

Factor $- 1$ out of $- 1 (4) - 1 (- x)$ .

$\frac{18 k (- 1 (4 - x)) (48 + x^{2})}{x^{3} {(12 - x)}^{3}}$

Step 6.4

Move the negative in front of the fraction.

$- \frac{((18 k) (4 - x)) (48 + x^{2})}{x^{3} {(12 - x)}^{3}}$