Pre-Algebra Examples

$\frac{{(5 + h)}^{3} - 125}{h}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $125$ as $5^{3}$ .

$\frac{{(5 + h)}^{3} - 5^{3}}{h}$

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 $a = 5 + h$ and $b = 5$ .

$\frac{(5 + h - 5) ({(5 + h)}^{2} + (5 + h) \cdot 5 + 5^{2})}{h}$

Step 1.3

Simplify.

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Step 1.3.1

Subtract $5$ from $5$ .

$\frac{(h + 0) ({(5 + h)}^{2} + (5 + h) \cdot 5 + 5^{2})}{h}$

Step 1.3.2

Add $h$ and $0$ .

$\frac{h ({(5 + h)}^{2} + (5 + h) \cdot 5 + 5^{2})}{h}$

Step 1.3.3

Rewrite ${(5 + h)}^{2}$ as $(5 + h) (5 + h)$ .

$\frac{h ((5 + h) (5 + h) + (5 + h) \cdot 5 + 5^{2})}{h}$

Step 1.3.4

Expand $(5 + h) (5 + h)$ using the FOIL Method.

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Step 1.3.4.1

Apply the distributive property.

$\frac{h (5 (5 + h) + h (5 + h) + (5 + h) \cdot 5 + 5^{2})}{h}$

Step 1.3.4.2

Apply the distributive property.

$\frac{h (5 \cdot 5 + 5 h + h (5 + h) + (5 + h) \cdot 5 + 5^{2})}{h}$

Step 1.3.4.3

Apply the distributive property.

$\frac{h (5 \cdot 5 + 5 h + h \cdot 5 + h \cdot h + (5 + h) \cdot 5 + 5^{2})}{h}$

Step 1.3.5

Simplify and combine like terms.

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Step 1.3.5.1

Simplify each term.

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Step 1.3.5.1.1

Multiply $5$ by $5$ .

$\frac{h (25 + 5 h + h \cdot 5 + h \cdot h + (5 + h) \cdot 5 + 5^{2})}{h}$

Step 1.3.5.1.2

Move $5$ to the left of $h$ .

$\frac{h (25 + 5 h + 5 \cdot h + h \cdot h + (5 + h) \cdot 5 + 5^{2})}{h}$

Step 1.3.5.1.3

Multiply $h$ by $h$ .

$\frac{h (25 + 5 h + 5 h + h^{2} + (5 + h) \cdot 5 + 5^{2})}{h}$

Step 1.3.5.2

Add $5 h$ and $5 h$ .

$\frac{h (25 + 10 h + h^{2} + (5 + h) \cdot 5 + 5^{2})}{h}$

Step 1.3.6

Apply the distributive property.

$\frac{h (25 + 10 h + h^{2} + 5 \cdot 5 + h \cdot 5 + 5^{2})}{h}$

Step 1.3.7

Multiply $5$ by $5$ .

$\frac{h (25 + 10 h + h^{2} + 25 + h \cdot 5 + 5^{2})}{h}$

Step 1.3.8

Move $5$ to the left of $h$ .

$\frac{h (25 + 10 h + h^{2} + 25 + 5 \cdot h + 5^{2})}{h}$

Step 1.3.9

Raise $5$ to the power of $2$ .

$\frac{h (25 + 10 h + h^{2} + 25 + 5 h + 25)}{h}$

Step 1.3.10

Add $25$ and $25$ .

$\frac{h (10 h + h^{2} + 50 + 5 h + 25)}{h}$

Step 1.3.11

Add $10 h$ and $5 h$ .

$\frac{h (15 h + h^{2} + 50 + 25)}{h}$

Step 1.3.12

Add $50$ and $25$ .

$\frac{h (15 h + h^{2} + 75)}{h}$

Step 1.3.13

Reorder terms.

$\frac{h (h^{2} + 15 h + 75)}{h}$

Step 2

Cancel the common factor of $h$ .

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Step 2.1

Cancel the common factor.

$\frac{h (h^{2} + 15 h + 75)}{h}$

Step 2.2

Divide $h^{2} + 15 h + 75$ by $1$ .

$h^{2} + 15 h + 75$