Pre-Algebra Examples

$(\frac{c - d}{c^{2} + c d} - \frac{c}{d^{2 + c d}}) \div (\frac{d^{2}}{c^{3} - c d^{2}} + \frac{1}{c + d})$

Step 1

Simplify and reorder the polynomial.

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Step 1.1

Rewrite the division as a fraction.

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

Step 1.2

Multiply the numerator and denominator of the fraction by $(c^{2} + c d) d^{2 + c d} (c^{3} - c d^{2}) (c + d)$ .

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Step 1.2.1

Multiply $\frac{\frac{c - d}{c^{2} + c d} - \frac{c}{d^{2 + c d}}}{\frac{d^{2}}{c^{3} - c d^{2}} + \frac{1}{c + d}}$ by $\frac{(c^{2} + c d) d^{2 + c d} (c^{3} - c d^{2}) (c + d)}{(c^{2} + c d) d^{2 + c d} (c^{3} - c d^{2}) (c + d)}$ .

$\frac{(c^{2} + c d) d^{2 + c d} (c^{3} - c d^{2}) (c + d)}{(c^{2} + c d) d^{2 + c d} (c^{3} - c d^{2}) (c + d)} \cdot \frac{\frac{c - d}{c^{2} + c d} - \frac{c}{d^{2 + c d}}}{\frac{d^{2}}{c^{3} - c d^{2}} + \frac{1}{c + d}}$

Step 1.2.2

Combine.

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

Step 1.3

Apply the distributive property.

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

Step 1.4

Simplify by cancelling.

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Step 1.4.1

Cancel the common factor of $c^{2} + c d$ .

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Step 1.4.1.1

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

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

Step 1.4.1.2

Cancel the common factor.

Step 1.4.1.3

Rewrite the expression.

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

Step 1.4.2

Cancel the common factor of $d^{2 + c d}$ .

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Step 1.4.2.1

Move the leading negative in $- \frac{c}{d^{2 + c d}}$ into the numerator.

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

Step 1.4.2.2

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

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

Step 1.4.2.3

Cancel the common factor.

Step 1.4.2.4

Rewrite the expression.

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

Step 1.4.3

Cancel the common factor of $c^{3} - c d^{2}$ .

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Step 1.4.3.1

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

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

Step 1.4.3.2

Cancel the common factor.

Step 1.4.3.3

Rewrite the expression.

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

Step 1.4.4

Multiply $d^{2 + c d}$ by $d^{2}$ by adding the exponents.

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Step 1.4.4.1

Move $d^{2}$ .

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

Step 1.4.4.2

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

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

Step 1.4.4.3

Add $2$ and $2$ .

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

Step 1.4.5

Cancel the common factor of $c + d$ .

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Step 1.4.5.1

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

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

Step 1.4.5.2

Cancel the common factor.

Step 1.4.5.3

Rewrite the expression.

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

Step 1.5

Simplify the numerator.

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Step 1.5.1

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

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Step 1.5.1.1

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

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

Step 1.5.1.2

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

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

Step 1.5.1.3

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

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

Step 1.5.2

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

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Step 1.5.2.1

Factor $c$ out of $c^{3}$ .

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

Step 1.5.2.2

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

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

Step 1.5.2.3

Factor $c$ out of $c \cdot c^{2} + c (- 1 d^{2})$ .

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

Step 1.5.3

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

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

Step 1.5.4

Combine exponents.

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Step 1.5.4.1

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

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

Step 1.5.4.2

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

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

Step 1.5.4.3

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

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

Step 1.5.4.4

Add $1$ and $1$ .

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

Step 1.5.5

Simplify each term.

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Step 1.5.5.1

Apply the distributive property.

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

Step 1.5.5.2

Rewrite using the commutative property of multiplication.

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

Step 1.5.5.3

Multiply $d^{2 + c d}$ by $d$ by adding the exponents.

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Step 1.5.5.3.1

Move $d$ .

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

Step 1.5.5.3.2

Multiply $d$ by $d^{2 + c d}$ .

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Step 1.5.5.3.2.1

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

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

Step 1.5.5.3.2.2

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

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

Step 1.5.5.3.3

Add $1$ and $2$ .

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

Step 1.5.5.4

Apply the distributive property.

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

Step 1.5.5.5

Rewrite using the commutative property of multiplication.

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

Step 1.5.5.6

Rewrite using the commutative property of multiplication.

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

Step 1.5.5.7

Simplify each term.

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Step 1.5.5.7.1

Multiply $c^{2}$ by $c$ by adding the exponents.

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Step 1.5.5.7.1.1

Move $c$ .

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

Step 1.5.5.7.1.2

Multiply $c$ by $c^{2}$ .

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Step 1.5.5.7.1.2.1

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

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

Step 1.5.5.7.1.2.2

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

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

Step 1.5.5.7.1.3

Add $1$ and $2$ .

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

Step 1.5.5.7.2

Multiply $c$ by $c$ by adding the exponents.

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Step 1.5.5.7.2.1

Move $c$ .

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

Step 1.5.5.7.2.2

Multiply $c$ by $c$ .

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

Step 1.6

Simplify the denominator.

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Step 1.6.1

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

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Step 1.6.1.1

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

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

Step 1.6.1.2

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

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

Step 1.6.1.3

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

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

Step 1.6.2

Factor $c$ out of $c^{2} + c d$ .

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Step 1.6.2.1

Factor $c$ out of $c^{2}$ .

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

Step 1.6.2.2

Factor $c$ out of $c d$ .

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

Step 1.6.2.3

Factor $c$ out of $c \cdot c + c (d)$ .

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

Step 1.6.3

Apply the distributive property.

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

Step 1.6.4

Multiply $d^{4 + c d}$ by $d$ by adding the exponents.

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Step 1.6.4.1

Multiply $d^{4 + c d}$ by $d$ .

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Step 1.6.4.1.1

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

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

Step 1.6.4.1.2

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

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

Step 1.6.4.2

Add $4$ and $1$ .

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

Step 1.6.5

Apply the distributive property.

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

Step 1.6.6

Rewrite using the commutative property of multiplication.

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

Step 1.6.7

Multiply $d^{2 + c d}$ by $d^{2}$ by adding the exponents.

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Step 1.6.7.1

Move $d^{2}$ .

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

Step 1.6.7.2

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

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

Step 1.6.7.3

Add $2$ and $2$ .

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

Step 1.6.8

Subtract $d^{4 + c d} c$ from $d^{4 + c d} c$ .

$\frac{c {(c + d)}^{2} (c - d) (d^{2 + c d} c - d^{3 + c d} - c^{3} - c^{2} d)}{c (c + d) (d^{c d + 5} + d^{2 + c d} c^{3} + 0)}$

Step 1.6.9

Add $d^{c d + 5} + d^{2 + c d} c^{3}$ and $0$ .

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

Step 1.7

Reduce the expression by cancelling the common factors.

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Step 1.7.1

Cancel the common factor of $c$ .

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Step 1.7.1.1

Cancel the common factor.

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

Step 1.7.1.2

Rewrite the expression.

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

Step 1.7.2

Cancel the common factor of ${(c + d)}^{2}$ and $c + d$ .

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Step 1.7.2.1

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

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

Step 1.7.2.2

Cancel the common factors.

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Step 1.7.2.2.1

Cancel the common factor.

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

Step 1.7.2.2.2

Rewrite the expression.

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

Step 1.7.3

Reorder factors in $\frac{(c + d) (c - d) (d^{2 + c d} c - d^{3 + c d} - c^{3} - c^{2} d)}{d^{c d + 5} + d^{2 + c d} c^{3}}$ .

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

Step 2

The degree cannot be determined because $\frac{(c + d) (c - d) (c d^{2 + c d} - d^{3 + c d} - c^{3} - c^{2} d)}{d^{c d + 5} + c^{3} d^{2 + c d}}$ is not a polynomial.

Not a polynomial