Algebra Examples

$(\frac{a^{2}}{a + n} - \frac{a^{3}}{a^{2} + n^{2} + 2 a n}) \div (\frac{a}{a + n} - \frac{a^{2}}{a^{2} - n^{2}})$

Step 1

Rewrite the division as a fraction.

$\frac{\frac{a^{2}}{a + n} - \frac{a^{3}}{a^{2} + n^{2} + 2 a n}}{\frac{a}{a + n} - \frac{a^{2}}{a^{2} - n^{2}}}$

Step 2

Multiply the numerator and denominator of the fraction by $(a + n) (a^{2} + n^{2} + 2 a n) (a^{2} - n^{2})$ .

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Step 2.1

Multiply $\frac{\frac{a^{2}}{a + n} - \frac{a^{3}}{a^{2} + n^{2} + 2 a n}}{\frac{a}{a + n} - \frac{a^{2}}{a^{2} - n^{2}}}$ by $\frac{(a + n) (a^{2} + n^{2} + 2 a n) (a^{2} - n^{2})}{(a + n) (a^{2} + n^{2} + 2 a n) (a^{2} - n^{2})}$ .

$\frac{(a + n) (a^{2} + n^{2} + 2 a n) (a^{2} - n^{2})}{(a + n) (a^{2} + n^{2} + 2 a n) (a^{2} - n^{2})} \cdot \frac{\frac{a^{2}}{a + n} - \frac{a^{3}}{a^{2} + n^{2} + 2 a n}}{\frac{a}{a + n} - \frac{a^{2}}{a^{2} - n^{2}}}$

Step 2.2

Combine.

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

Step 3

Apply the distributive property.

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

Step 4

Simplify by cancelling.

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Step 4.1

Cancel the common factor of $a + n$ .

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Step 4.1.1

Factor $a + n$ out of $(a + n) (a^{2} + n^{2} + 2 a n) (a^{2} - n^{2})$ .

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

Step 4.1.2

Cancel the common factor.

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $a^{2} + n^{2} + 2 a n$ .

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Step 4.2.1

Move the leading negative in $- \frac{a^{3}}{a^{2} + n^{2} + 2 a n}$ into the numerator.

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

Step 4.2.2

Factor $a^{2} + n^{2} + 2 a n$ out of $(a + n) (a^{2} + n^{2} + 2 a n) (a^{2} - n^{2})$ .

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

Step 4.2.3

Cancel the common factor.

Step 4.2.4

Rewrite the expression.

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

Step 4.3

Cancel the common factor of $a + n$ .

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Step 4.3.1

Factor $a + n$ out of $(a + n) (a^{2} + n^{2} + 2 a n) (a^{2} - n^{2})$ .

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

Step 4.3.2

Cancel the common factor.

Step 4.3.3

Rewrite the expression.

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

Step 4.4

Cancel the common factor of $a^{2} - n^{2}$ .

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Step 4.4.1

Move the leading negative in $- \frac{a^{2}}{a^{2} - n^{2}}$ into the numerator.

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

Step 4.4.2

Factor $a^{2} - n^{2}$ out of $(a + n) (a^{2} + n^{2} + 2 a n) (a^{2} - n^{2})$ .

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

Step 4.4.3

Cancel the common factor.

Step 4.4.4

Rewrite the expression.

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

Step 5

Simplify the numerator.

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Step 5.1

Factor $(a^{2} - n^{2}) a^{2}$ out of $(a^{2} + n^{2} + 2 a n) (a^{2} - n^{2}) a^{2} + (a + n) (a^{2} - n^{2}) (- a^{3})$ .

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Step 5.1.1

Factor $(a^{2} - n^{2}) a^{2}$ out of $(a^{2} + n^{2} + 2 a n) (a^{2} - n^{2}) a^{2}$ .

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

Step 5.1.2

Factor $(a^{2} - n^{2}) a^{2}$ out of $(a + n) (a^{2} - n^{2}) (- a^{3})$ .

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

Step 5.1.3

Factor $(a^{2} - n^{2}) a^{2}$ out of $(a^{2} - n^{2}) a^{2} (a^{2} + n^{2} + 2 a n) + (a^{2} - n^{2}) a^{2} ((a + n) (- a))$ .

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

Step 5.2

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

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Rewrite using the commutative property of multiplication.

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

Step 5.5

Rewrite using the commutative property of multiplication.

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

Step 5.6

Multiply $a$ by $a$ by adding the exponents.

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Step 5.6.1

Move $a$ .

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

Step 5.6.2

Multiply $a$ by $a$ .

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

Step 5.7

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

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

Step 5.8

Add $n^{2}$ and $0$ .

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

Step 5.9

Subtract $n a$ from $2 a n$ .

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Step 5.9.1

Move $n$ .

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

Step 5.9.2

Subtract $a n$ from $2 a n$ .

$\frac{(a + n) (a - n) a^{2} (n^{2} + 1 a n)}{(a^{2} + n^{2} + 2 a n) (a^{2} - n^{2}) a + (a + n) (a^{2} + n^{2} + 2 a n) (- a^{2})}$

Step 5.10

Rewrite $n^{2} + 1 a n$ in a factored form.

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Step 5.10.1

Factor $n$ out of $n^{2} + 1 a n$ .

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Step 5.10.1.1

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

$\frac{(a + n) (a - n) a^{2} (n \cdot n + 1 a n)}{(a^{2} + n^{2} + 2 a n) (a^{2} - n^{2}) a + (a + n) (a^{2} + n^{2} + 2 a n) (- a^{2})}$

Step 5.10.1.2

Factor $n$ out of $1 a n$ .

$\frac{(a + n) (a - n) a^{2} (n \cdot n + n (1 a))}{(a^{2} + n^{2} + 2 a n) (a^{2} - n^{2}) a + (a + n) (a^{2} + n^{2} + 2 a n) (- a^{2})}$

Step 5.10.1.3

Factor $n$ out of $n \cdot n + n (1 a)$ .

$\frac{(a + n) (a - n) a^{2} (n (n + 1 a))}{(a^{2} + n^{2} + 2 a n) (a^{2} - n^{2}) a + (a + n) (a^{2} + n^{2} + 2 a n) (- a^{2})}$

Step 5.10.2

Multiply $a$ by $1$ .

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

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

Step 5.11

Combine exponents.

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Step 5.11.1

Reorder terms.

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

Step 5.11.2

Raise $a + n$ to the power of $1$ .

$\frac{(a - n) a^{2} n ({(a + n)}^{1} (a + n))}{(a^{2} + n^{2} + 2 a n) (a^{2} - n^{2}) a + (a + n) (a^{2} + n^{2} + 2 a n) (- a^{2})}$

Step 5.11.3

Raise $a + n$ to the power of $1$ .

$\frac{(a - n) a^{2} n ({(a + n)}^{1} {(a + n)}^{1})}{(a^{2} + n^{2} + 2 a n) (a^{2} - n^{2}) a + (a + n) (a^{2} + n^{2} + 2 a n) (- a^{2})}$

Step 5.11.4

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

$\frac{(a - n) a^{2} n {(a + n)}^{1 + 1}}{(a^{2} + n^{2} + 2 a n) (a^{2} - n^{2}) a + (a + n) (a^{2} + n^{2} + 2 a n) (- a^{2})}$

Step 5.11.5

Add $1$ and $1$ .

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

Step 6

Simplify the denominator.

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Step 6.1

Factor $(a^{2} + n^{2} + 2 a n) a$ out of $(a^{2} + n^{2} + 2 a n) (a^{2} - n^{2}) a + (a + n) (a^{2} + n^{2} + 2 a n) (- a^{2})$ .

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Step 6.1.1

Factor $(a^{2} + n^{2} + 2 a n) a$ out of $(a^{2} + n^{2} + 2 a n) (a^{2} - n^{2}) a$ .

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

Step 6.1.2

Factor $(a^{2} + n^{2} + 2 a n) a$ out of $(a + n) (a^{2} + n^{2} + 2 a n) (- a^{2})$ .

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

Step 6.1.3

Factor $(a^{2} + n^{2} + 2 a n) a$ out of $(a^{2} + n^{2} + 2 a n) a (a^{2} - n^{2}) + (a^{2} + n^{2} + 2 a n) a ((a + n) (- a))$ .

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

Step 6.2

Factor using the perfect square rule.

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Step 6.2.1

Rearrange terms.

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

Step 6.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a n = 2 \cdot a \cdot n$

Step 6.2.3

Rewrite the polynomial.

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

Step 6.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = a$ and $b = n$ .

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Rewrite using the commutative property of multiplication.

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

Step 6.5

Rewrite using the commutative property of multiplication.

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

Step 6.6

Multiply $a$ by $a$ by adding the exponents.

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Step 6.6.1

Move $a$ .

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

Step 6.6.2

Multiply $a$ by $a$ .

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

Step 6.7

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

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

Step 6.8

Add $- n^{2}$ and $0$ .

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

Step 6.9

Factor $n$ out of $- n^{2} - n a$ .

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Step 6.9.1

Factor $n$ out of $- n^{2}$ .

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

Step 6.9.2

Factor $n$ out of $- n a$ .

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

Step 6.9.3

Factor $n$ out of $n (- n) + n (- a)$ .

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

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

Step 6.10

Combine exponents.

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Step 6.10.1

Factor $- 1$ out of $a$ .

$\frac{(a - n) a^{2} n {(a + n)}^{2}}{{(- 1 (- a) + n)}^{2} a n (- n - a)}$

Step 6.10.2

Factor $- 1$ out of $n$ .

$\frac{(a - n) a^{2} n {(a + n)}^{2}}{{(- 1 (- a) - 1 (- n))}^{2} a n (- n - a)}$

Step 6.10.3

Factor $- 1$ out of $- 1 (- a) - 1 (- n)$ .

$\frac{(a - n) a^{2} n {(a + n)}^{2}}{{(- 1 (- a - n))}^{2} a n (- n - a)}$

Step 6.10.4

Apply the product rule to $- 1 (- a - n)$ .

$\frac{(a - n) a^{2} n {(a + n)}^{2}}{{(- 1)}^{2} {(- a - n)}^{2} a n (- n - a)}$

Step 6.10.5

Raise $- 1$ to the power of $2$ .

$\frac{(a - n) a^{2} n {(a + n)}^{2}}{1 {(- a - n)}^{2} a n (- n - a)}$

Step 6.10.6

Multiply ${(- a - n)}^{2}$ by $1$ .

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

Step 6.10.7

Reorder terms.

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

Step 6.10.8

Raise $- a - n$ to the power of $1$ .

$\frac{(a - n) a^{2} n {(a + n)}^{2}}{a n ({(- a - n)}^{2} {(- a - n)}^{1})}$

Step 6.10.9

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

$\frac{(a - n) a^{2} n {(a + n)}^{2}}{a n {(- a - n)}^{2 + 1}}$

Step 6.10.10

Add $2$ and $1$ .

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

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $a^{2}$ and $a$ .

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Step 7.1.1

Factor $a$ out of $(a - n) a^{2} n {(a + n)}^{2}$ .

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

Step 7.1.2

Cancel the common factors.

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Step 7.1.2.1

Factor $a$ out of $a n {(- a - n)}^{3}$ .

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

Step 7.1.2.2

Cancel the common factor.

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

Step 7.1.2.3

Rewrite the expression.

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

Step 7.2

Cancel the common factor of $n$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

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

Step 7.3

Cancel the common factor of ${(a + n)}^{2}$ and ${(- a - n)}^{3}$ .

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Step 7.3.1

Factor $- 1$ out of $a$ .

$\frac{(a - n) a {(- 1 (- a) + n)}^{2}}{{(- a - n)}^{3}}$

Step 7.3.2

Factor $- 1$ out of $n$ .

$\frac{(a - n) a {(- 1 (- a) - 1 (- n))}^{2}}{{(- a - n)}^{3}}$

Step 7.3.3

Factor $- 1$ out of $- 1 (- a) - 1 (- n)$ .

$\frac{(a - n) a {(- 1 (- a - n))}^{2}}{{(- a - n)}^{3}}$

Step 7.3.4

Apply the product rule to $- 1 (- a - n)$ .

$\frac{(a - n) a ({(- 1)}^{2} {(- a - n)}^{2})}{{(- a - n)}^{3}}$

Step 7.3.5

Raise $- 1$ to the power of $2$ .

$\frac{(a - n) a (1 {(- a - n)}^{2})}{{(- a - n)}^{3}}$

Step 7.3.6

Multiply ${(- a - n)}^{2}$ by $1$ .

$\frac{(a - n) a {(- a - n)}^{2}}{{(- a - n)}^{3}}$

Step 7.3.7

Factor ${(- a - n)}^{2}$ out of $(a - n) a {(- a - n)}^{2}$ .

$\frac{{(- a - n)}^{2} ((a - n) a)}{{(- a - n)}^{3}}$

Step 7.3.8

Cancel the common factors.

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Step 7.3.8.1

Factor ${(- a - n)}^{2}$ out of ${(- a - n)}^{3}$ .

$\frac{{(- a - n)}^{2} ((a - n) a)}{{(- a - n)}^{2} (- a - n)}$

Step 7.3.8.2

Cancel the common factor.

$\frac{{(- a - n)}^{2} ((a - n) a)}{{(- a - n)}^{2} (- a - n)}$

Step 7.3.8.3

Rewrite the expression.

$\frac{(a - n) a}{- a - n}$

Step 7.4

Factor $- 1$ out of $- a$ .

$\frac{(a - n) a}{- (a) - n}$

Step 7.5

Factor $- 1$ out of $- n$ .

$\frac{(a - n) a}{- (a) - (n)}$

Step 7.6

Factor $- 1$ out of $- (a) - (n)$ .

$\frac{(a - n) a}{- (a + n)}$

Step 7.7

Rewrite negatives.

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Step 7.7.1

Rewrite $- (a + n)$ as $- 1 (a + n)$ .

$\frac{(a - n) a}{- 1 (a + n)}$

Step 7.7.2

Move the negative in front of the fraction.

$- \frac{(a - n) a}{a + n}$

Step 7.7.3

Reorder factors in $- \frac{(a - n) a}{a + n}$ .

$- \frac{a (a - n)}{a + n}$