Algebra Examples

$\frac{m - 3 n}{m^{3} - n^{3}} - \frac{2 n}{n^{3} - m^{3}}$

Step 1

Simplify each term.

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Step 1.1

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 $a = m$ and $b = n$ .

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

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 $a = n$ and $b = m$ .

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

Step 2

Simplify with factoring out.

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Step 2.1

Factor $- 1$ out of $n$ .

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

Step 2.2

Factor $- 1$ out of $- m$ .

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

Step 2.3

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

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

Step 2.4

Reorder.

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Step 2.4.1

Reorder terms.

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

Step 2.4.2

Reorder terms.

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

Step 2.4.3

Reorder terms.

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

Step 3

To write $\frac{m - 3 n}{(m - n) (m^{2} + n^{2} + m n)}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

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

Step 4

Write each expression with a common denominator of $(m - n) (m^{2} + n^{2} + m n) \cdot - 1$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

Reorder the factors of $(m - n) (m^{2} + n^{2} + m n) \cdot - 1$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Move $- 1$ to the left of $m$ .

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

Step 6.3

Multiply $- 1$ by $- 3$ .

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

Step 6.4

Rewrite $- 1 m$ as $- m$ .

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

Step 6.5

Subtract $2 n$ from $3 n$ .

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

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Cancel the common factor of $- m + n$ and $m - n$ .

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Step 7.1.1

Factor $- 1$ out of $- m$ .

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

Step 7.1.2

Factor $- 1$ out of $n$ .

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

Step 7.1.3

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

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

Step 7.1.4

Rewrite $- (m - n)$ as $- 1 (m - n)$ .

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

Step 7.1.5

Cancel the common factor.

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

Step 7.1.6

Rewrite the expression.

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

Step 7.2

Dividing two negative values results in a positive value.

$\frac{1}{m^{2} + n^{2} + m n}$