Basic Math Examples

$\frac{4 a^{3} - 4 {(a + h)}^{3}}{h}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $4$ out of $4 a^{3} - 4 {(a + h)}^{3}$ .

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Step 1.1.1

Factor $4$ out of $4 a^{3}$ .

$\frac{4 (a^{3}) - 4 {(a + h)}^{3}}{h}$

Step 1.1.2

Factor $4$ out of $- 4 {(a + h)}^{3}$ .

$\frac{4 (a^{3}) + 4 (- {(a + h)}^{3})}{h}$

Step 1.1.3

Factor $4$ out of $4 (a^{3}) + 4 (- {(a + h)}^{3})$ .

$\frac{4 (a^{3} - {(a + h)}^{3})}{h}$

Step 1.2

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

$\frac{4 ((a - (a + h)) (a^{2} + a (a + h) + {(a + h)}^{2}))}{h}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$\frac{4 ((a - a - h) (a^{2} + a (a + h) + {(a + h)}^{2}))}{h}$

Step 1.3.2

Subtract $a$ from $a$ .

$\frac{4 ((0 - h) (a^{2} + a (a + h) + {(a + h)}^{2}))}{h}$

Step 1.3.3

Subtract $h$ from $0$ .

$\frac{4 (- h (a^{2} + a (a + h) + {(a + h)}^{2}))}{h}$

Step 1.3.4

Apply the distributive property.

$\frac{4 (- h (a^{2} + a \cdot a + a h + {(a + h)}^{2}))}{h}$

Step 1.3.5

Multiply $a$ by $a$ .

$\frac{4 (- h (a^{2} + a^{2} + a h + {(a + h)}^{2}))}{h}$

Step 1.3.6

Rewrite ${(a + h)}^{2}$ as $(a + h) (a + h)$ .

$\frac{4 (- h (a^{2} + a^{2} + a h + (a + h) (a + h)))}{h}$

Step 1.3.7

Expand $(a + h) (a + h)$ using the FOIL Method.

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Step 1.3.7.1

Apply the distributive property.

$\frac{4 (- h (a^{2} + a^{2} + a h + a (a + h) + h (a + h)))}{h}$

Step 1.3.7.2

Apply the distributive property.

$\frac{4 (- h (a^{2} + a^{2} + a h + a \cdot a + a h + h (a + h)))}{h}$

Step 1.3.7.3

Apply the distributive property.

$\frac{4 (- h (a^{2} + a^{2} + a h + a \cdot a + a h + h a + h \cdot h))}{h}$

Step 1.3.8

Simplify and combine like terms.

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Step 1.3.8.1

Simplify each term.

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Step 1.3.8.1.1

Multiply $a$ by $a$ .

$\frac{4 (- h (a^{2} + a^{2} + a h + a^{2} + a h + h a + h \cdot h))}{h}$

Step 1.3.8.1.2

Multiply $h$ by $h$ .

$\frac{4 (- h (a^{2} + a^{2} + a h + a^{2} + a h + h a + h^{2}))}{h}$

Step 1.3.8.2

Add $a h$ and $h a$ .

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Step 1.3.8.2.1

Reorder $h$ and $a$ .

$\frac{4 (- h (a^{2} + a^{2} + a h + a^{2} + a h + a h + h^{2}))}{h}$

Step 1.3.8.2.2

Add $a h$ and $a h$ .

$\frac{4 (- h (a^{2} + a^{2} + a h + a^{2} + 2 a h + h^{2}))}{h}$

Step 1.3.9

Add $a^{2}$ and $a^{2}$ .

$\frac{4 (- h (2 a^{2} + a h + a^{2} + 2 a h + h^{2}))}{h}$

Step 1.3.10

Add $2 a^{2}$ and $a^{2}$ .

$\frac{4 (- h (3 a^{2} + a h + 2 a h + h^{2}))}{h}$

Step 1.3.11

Add $a h$ and $2 a h$ .

$\frac{4 (- h (3 a^{2} + 3 a h + h^{2}))}{h}$

$\frac{4 \cdot - 1 h (3 a^{2} + 3 a h + h^{2})}{h}$

Step 1.4

Combine exponents.

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Step 1.4.1

Factor out negative.

$\frac{- (4 h (3 a^{2} + 3 a h + h^{2}))}{h}$

Step 1.4.2

Multiply $4$ by $- 1$ .

$\frac{- 4 (h (3 a^{2} + 3 a h + h^{2}))}{h}$

$\frac{- 4 h (3 a^{2} + 3 a h + h^{2})}{h}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $h$ .

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Step 2.1.1

Cancel the common factor.

$\frac{- 4 h (3 a^{2} + 3 a h + h^{2})}{h}$

Step 2.1.2

Divide $- 4 (3 a^{2} + 3 a h + h^{2})$ by $1$ .

$- 4 (3 a^{2} + 3 a h + h^{2})$

Step 2.2

Apply the distributive property.

$- 4 (3 a^{2}) - 4 (3 a h) - 4 h^{2}$

Step 3

Simplify.

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Step 3.1

Multiply $3$ by $- 4$ .

$- 12 a^{2} - 4 (3 a h) - 4 h^{2}$

Step 3.2

Multiply $3$ by $- 4$ .

$- 12 a^{2} - 12 (a h) - 4 h^{2}$

$- 12 a^{2} - 12 a h - 4 h^{2}$