Calculus Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $m - n$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Divide $m + n$ by $1$ .

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

Step 3.2

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

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Step 3.2.1

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

$(m + n) \frac{m \cdot m - n m}{m}$

Step 3.2.2

Factor $m$ out of $- n m$ .

$(m + n) \frac{m \cdot m + m (- n)}{m}$

Step 3.2.3

Factor $m$ out of $m \cdot m + m (- n)$ .

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

Step 3.3

Cancel the common factor of $m$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Divide $m - n$ by $1$ .

$(m + n) (m - n)$

Step 4

Expand $(m + n) (m - n)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$m (m - n) + n (m - n)$

Step 4.2

Apply the distributive property.

$m \cdot m + m (- n) + n (m - n)$

Step 4.3

Apply the distributive property.

$m \cdot m + m (- n) + n m + n (- n)$

Step 5

Simplify terms.

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Step 5.1

Combine the opposite terms in $m \cdot m + m (- n) + n m + n (- n)$ .

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Step 5.1.1

Reorder the factors in the terms $m (- n)$ and $n m$ .

$m \cdot m - m n + m n + n (- n)$

Step 5.1.2

Add $- m n$ and $m n$ .

$m \cdot m + 0 + n (- n)$

Step 5.1.3

Add $m \cdot m$ and $0$ .

$m \cdot m + n (- n)$

Step 5.2

Simplify each term.

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Step 5.2.1

Multiply $m$ by $m$ .

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

Step 5.2.2

Rewrite using the commutative property of multiplication.

$m^{2} - n \cdot n$

Step 5.2.3

Multiply $n$ by $n$ by adding the exponents.

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Step 5.2.3.1

Move $n$ .

$m^{2} - (n \cdot n)$

Step 5.2.3.2

Multiply $n$ by $n$ .

$m^{2} - n^{2}$