Pre-Algebra Examples

$\frac{2 c^{3} d + 4 c^{2} d^{2} - 12 c d^{- 3}}{2 c d}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $2 c d^{- 3}$ out of $2 c^{3} d + 4 c^{2} d^{2} - 12 c d^{- 3}$ .

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Step 1.1.1

Factor $2 c d^{- 3}$ out of $2 c^{3} d$ .

$\frac{2 c d^{- 3} (c^{2} d^{4}) + 4 c^{2} d^{2} - 12 c d^{- 3}}{2 c d}$

Step 1.1.2

Factor $2 c d^{- 3}$ out of $4 c^{2} d^{2}$ .

$\frac{2 c d^{- 3} (c^{2} d^{4}) + 2 c d^{- 3} (2 c d^{5}) - 12 c d^{- 3}}{2 c d}$

Step 1.1.3

Factor $2 c d^{- 3}$ out of $- 12 c d^{- 3}$ .

$\frac{2 c d^{- 3} (c^{2} d^{4}) + 2 c d^{- 3} (2 c d^{5}) + 2 c d^{- 3} (- 6)}{2 c d}$

Step 1.1.4

Factor $2 c d^{- 3}$ out of $2 c d^{- 3} (c^{2} d^{4}) + 2 c d^{- 3} (2 c d^{5})$ .

$\frac{2 c d^{- 3} (c^{2} d^{4} + 2 c d^{5}) + 2 c d^{- 3} (- 6)}{2 c d}$

Step 1.1.5

Factor $2 c d^{- 3}$ out of $2 c d^{- 3} (c^{2} d^{4} + 2 c d^{5}) + 2 c d^{- 3} (- 6)$ .

$\frac{2 c d^{- 3} (c^{2} d^{4} + 2 c d^{5} - 6)}{2 c d}$

Step 1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2 c \frac{1}{d^{3}} (c^{2} d^{4} + 2 c d^{5} - 6)}{2 c d}$

Step 1.3

Combine exponents.

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Step 1.3.1

Combine $2$ and $\frac{1}{d^{3}}$ .

$\frac{c \frac{2}{d^{3}} (c^{2} d^{4} + 2 c d^{5} - 6)}{2 c d}$

Step 1.3.2

Combine $c$ and $\frac{2}{d^{3}}$ .

$\frac{\frac{c \cdot 2}{d^{3}} (c^{2} d^{4} + 2 c d^{5} - 6)}{2 c d}$

Step 1.4

Move $2$ to the left of $c$ .

$\frac{\frac{2 c}{d^{3}} (c^{2} d^{4} + 2 c d^{5} - 6)}{2 c d}$

Step 2

Factor $\frac{2 c}{d^{3}}$ out of $\frac{\frac{2 c}{d^{3}} (c^{2} d^{4} + 2 c d^{5} - 6)}{2 c d}$ .

$\frac{2 c}{d^{3}} \cdot \frac{c^{2} d^{4} + 2 c d^{5} - 6}{2 c d}$

Step 3

Cancel the common factor of $2 c$ .

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Step 3.1

Factor $2 c$ out of $2 c d$ .

$\frac{2 c}{d^{3}} \cdot \frac{c^{2} d^{4} + 2 c d^{5} - 6}{2 c (d)}$

Step 3.2

Cancel the common factor.

$\frac{2 c}{d^{3}} \cdot \frac{c^{2} d^{4} + 2 c d^{5} - 6}{2 c d}$

Step 3.3

Rewrite the expression.

$\frac{1}{d^{3}} \cdot \frac{c^{2} d^{4} + 2 c d^{5} - 6}{d}$

Step 4

Multiply $\frac{1}{d^{3}}$ by $\frac{c^{2} d^{4} + 2 c d^{5} - 6}{d}$ .

$\frac{c^{2} d^{4} + 2 c d^{5} - 6}{d^{3} d}$

Step 5

Multiply $d^{3}$ by $d$ by adding the exponents.

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Step 5.1

Multiply $d^{3}$ by $d$ .

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Step 5.1.1

Raise $d$ to the power of $1$ .

$\frac{c^{2} d^{4} + 2 c d^{5} - 6}{d^{3} d^{1}}$

Step 5.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{c^{2} d^{4} + 2 c d^{5} - 6}{d^{3 + 1}}$

Step 5.2

Add $3$ and $1$ .

$\frac{c^{2} d^{4} + 2 c d^{5} - 6}{d^{4}}$

Step 6

Split the fraction $\frac{c^{2} d^{4} + 2 c d^{5} - 6}{d^{4}}$ into two fractions.

$\frac{c^{2} d^{4} + 2 c d^{5}}{d^{4}} + \frac{- 6}{d^{4}}$

Step 7

Split the fraction $\frac{c^{2} d^{4} + 2 c d^{5}}{d^{4}}$ into two fractions.

$\frac{c^{2} d^{4}}{d^{4}} + \frac{2 c d^{5}}{d^{4}} + \frac{- 6}{d^{4}}$

Step 8

Cancel the common factor of $d^{4}$ .

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Step 8.1

Cancel the common factor.

$\frac{c^{2} d^{4}}{d^{4}} + \frac{2 c d^{5}}{d^{4}} + \frac{- 6}{d^{4}}$

Step 8.2

Divide $c^{2}$ by $1$ .

$c^{2} + \frac{2 c d^{5}}{d^{4}} + \frac{- 6}{d^{4}}$

Step 9

Cancel the common factor of $d^{5}$ and $d^{4}$ .

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Step 9.1

Factor $d^{4}$ out of $2 c d^{5}$ .

$c^{2} + \frac{d^{4} (2 c d)}{d^{4}} + \frac{- 6}{d^{4}}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Multiply by $1$ .

$c^{2} + \frac{d^{4} (2 c d)}{d^{4} \cdot 1} + \frac{- 6}{d^{4}}$

Step 9.2.2

Cancel the common factor.

$c^{2} + \frac{d^{4} (2 c d)}{d^{4} \cdot 1} + \frac{- 6}{d^{4}}$

Step 9.2.3

Rewrite the expression.

$c^{2} + \frac{2 c d}{1} + \frac{- 6}{d^{4}}$

Step 9.2.4

Divide $2 c d$ by $1$ .

$c^{2} + 2 c d + \frac{- 6}{d^{4}}$

Step 10

Move the negative in front of the fraction.

$c^{2} + 2 c d - \frac{6}{d^{4}}$